@article{briceno-arias_combettes_silva_2024, title={Perspective functions with nonlinear scaling}, volume={2}, ISSN={["1793-6683"]}, DOI={10.1142/S0219199723500657}, abstractNote={ The classical perspective of a function is a construction which transforms a convex function into one that is jointly convex with respect to an auxiliary scaling variable. Motivated by applications in several areas of applied analysis, we investigate an extension of this construct in which the scaling variable is replaced by a nonlinear term. Our construction is placed in the general context of locally convex spaces and it generates a lower semicontinuous convex function under broad assumptions on the underlying functions. Various convex-analytical properties are established and closed-form expressions are derived. Several applications are presented. }, journal={COMMUNICATIONS IN CONTEMPORARY MATHEMATICS}, author={Briceno-Arias, Luis M. and Combettes, Patrick L. and Silva, Francisco J.}, year={2024}, month={Feb} }
 @article{the geometry of monotone operator  splitting methods (to appear, available on arxiv)_2024, journal={Acta Numerica}, year={2024} }
 @article{briceno-arias_combettes_2023, title={A Perturbation Framework for Convex Minimization and Monotone Inclusion Problems with Nonlinear Compositions}, volume={10}, ISSN={["1526-5471"]}, DOI={10.1287/moor.2022.0180}, abstractNote={ We introduce a framework based on Rockafellar’s perturbation theory to analyze and solve general nonsmooth convex minimization and monotone inclusion problems involving nonlinearly composed functions as well as linear compositions. Such problems have been investigated only from a primal perspective and only for nonlinear compositions of smooth functions in finite-dimensional spaces in the absence of linear compositions. In the context of Banach spaces, the proposed perturbation analysis serves as a foundation for the construction of a dual problem and of a maximally monotone Kuhn–Tucker operator, which is decomposable as the sum of simpler monotone operators. In the Hilbertian setting, this decomposition leads to a block-iterative primal-dual algorithm that fully splits all the components of the problem and appears to be the first proximal splitting algorithm for handling nonlinear composite problems. Various applications are discussed. }, journal={MATHEMATICS OF OPERATIONS RESEARCH}, author={Briceno-Arias, Luis M. and Combettes, Patrick L.}, year={2023}, month={Oct} }
 @article{combettes_2023, title={Resolvent and Proximal Compositions}, volume={31}, ISSN={["1877-0541"]}, DOI={10.1007/s11228-023-00678-z}, abstractNote={We introduce the resolvent composition, a monotonicity-preserving operation between a linear operator and a set-valued operator, as well as the proximal composition, a convexity-preserving operation between a linear operator and a function. The two operations are linked by the fact that, under mild assumptions, the subdifferential of the proximal composition of a convex function is the resolvent composition of its subdifferential. The resolvent and proximal compositions are shown to encapsulate known concepts, such as the resolvent and proximal averages, as well as new operations pertinent to the analysis of equilibrium problems. A large core of properties of these compositions is established and several instantiations are discussed. Applications to the relaxation of monotone inclusion and convex optimization problems are presented.}, number={3}, journal={SET-VALUED AND VARIATIONAL ANALYSIS}, author={Combettes, Patrick L.}, year={2023}, month={Sep} }
 @article{combettes_woodstock_2022, title={A Variational Inequality Model for the Construction of Signals from Inconsistent Nonlinear Equations\ast}, volume={15}, ISSN={["1936-4954"]}, DOI={10.1137/21M1420368}, abstractNote={Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies a number of nonlinear equations involving firmly nonexpansive operators. We investigate this formalism in the case when, due to inaccurate modeling or perturbations, the nonlinear equations are inconsistent. A relaxed formulation of the original problem is proposed in the form of a variational inequality. The properties of the relaxed problem are investigated and a provenly convergent block-iterative algorithm, whereby only blocks of the underlying firmly nonexpansive operators are activated at a given iteration, is devised to solve it. Numerical experiments illustrate robust recoveries in several signal and image processing applications.}, number={1}, journal={SIAM JOURNAL ON IMAGING SCIENCES}, author={Combettes, Patrick L. and Woodstock, Zev C.}, year={2022}, pages={84–109} }
 @article{bui_combettes_woodstock_2022, title={BLOCK-ACTIVATED ALGORITHMS FOR MULTICOMPONENT FULLY NONSMOOTH MINIMIZATION}, ISSN={["1520-6149"]}, DOI={10.1109/ICASSP43922.2022.9747479}, abstractNote={Under consideration are multicomponent minimization problems involving a separable nonsmooth convex function penalizing the components individually, and nonsmooth convex coupling terms penalizing linear mixtures of the components. We investigate the application of block-activated proximal algorithms for solving such problems, i.e., algorithms which, at each iteration, need to use only a block of the underlying functions, as opposed to all of them as in standard methods. For smooth coupling functions, several block-activated algorithms exist and they are well understood. By contrast, in the fully nonsmooth case, few block-activated methods are available and little effort has been devoted to assessing them. Our goal is to shed more light on the implementation, the features, and the behavior of these algorithms, compare their merits, and provide machine learning and image recovery experiments illustrating their performance.}, journal={2022 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH AND SIGNAL PROCESSING (ICASSP)}, author={Bui, Minh N. and Combettes, Patrick L. and Woodstock, Zev C.}, year={2022}, pages={5428–5432} }
 @article{combettes_woodstock_2022, title={SIGNAL RECOVERY FROM INCONSISTENT NONLINEAR OBSERVATIONS}, ISSN={["1520-6149"]}, DOI={10.1109/ICASSP43922.2022.9746145}, abstractNote={We show that many nonlinear observation models in signal recovery can be represented using firmly nonexpansive operators. To address problems with inaccurate measurements, we propose solving a variational inequality relaxation which is guaranteed to possess solutions under mild conditions and which coincides with the original problem if it happens to be consistent. We then present an efficient algorithm for its solution, as well as numerical applications in signal and image recovery, including an experimental operator-theoretic method of promoting sparsity.}, journal={2022 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH AND SIGNAL PROCESSING (ICASSP)}, author={Combettes, Patrick L. and Woodstock, Zev C.}, year={2022}, pages={5872–5876} }
 @inproceedings{combettes_woodstock_2021, place={Hoboken, New Jersey}, title={A Fixed Point Framework for Recovering Signals from Nonlinear Transformations}, url={http://dx.doi.org/10.23919/eusipco47968.2020.9287736}, DOI={10.23919/eusipco47968.2020.9287736}, abstractNote={We consider the problem of recovering a signal from nonlinear transformations, under convex constraints modeling a priori information. Standard feasibility and optimization methods are ill-suited to tackle this problem due to the nonlinearities. We show that, in many common applications, the transformation model can be associated with fixed point equations involving firmly nonexpansive operators. In turn, the recovery problem is reduced to a tractable common fixed point formulation, which is solved efficiently by a provably convergent, block-iterative algorithm. Applications to signal and image recovery are demonstrated. Inconsistent problems are also addressed.}, booktitle={2020 28th European Signal Processing Conference (EUSIPCO)}, publisher={IEEE}, author={Combettes, Patrick L. and Woodstock, Zev C.}, year={2021}, month={Jan}, pages={2120–2124} }
 @book{bùi_combettes_2021, title={Analysis and numerical solution of a modular convex Nash equilibrium problem}, url={https://hal.archives-ouvertes.fr/hal-03412172}, number={hal-03412172}, author={Bùi, M.N. and Combettes, P.L.}, year={2021} }
 @book{bùi_combettes_woodstock_2021, title={Block-Activated Algorithms for Multicomponent Fully Nonsmooth Minimization}, url={https://arxiv.org/abs/2103.00520}, DOI={10.48550/arXiv.2103.00520}, abstractNote={Under consideration are multicomponent minimization problems involving a separable nonsmooth convex function penalizing the components individually, and nonsmooth convex coupling terms penalizing linear mixtures of the components. We investigate block-activated proximal algorithms for solving such problems, i.e., algorithms which, at each iteration, need to use only a block of the underlying functions, as opposed to all of them as in standard methods. For smooth coupling functions, several block-activated algorithms exist and they are well understood. By contrast, in the fully nonsmooth case, few block-activated methods are available and little effort has been devoted to assessing them. Our goal is to shed more light on the implementation, the features, and the behavior of these algorithms, compare their merits, and provide machine learning and image recovery experiments illustrating their performance.}, number={2103.005202103.00520}, institution={arXiv}, author={Bùi, Minh N. and Combettes, Patrick L. and Woodstock, Zev C.}, year={2021} }
 @article{bui_combettes_2021, title={Bregman Forward-Backward Operator Splitting}, volume={29}, ISBN={1877-0541}, url={https://doi.org/10.1007/s11228-020-00563-z}, DOI={10.1007/s11228-020-00563-z}, abstractNote={We establish the convergence of the forward-backward splitting algorithm based on Bregman distances for the sum of two monotone operators in reflexive Banach spaces. Even in Euclidean spaces, the convergence of this algorithm has so far been proved only in the case of minimization problems. The proposed framework features Bregman distances that vary over the iterations and a novel assumption on the single-valued operator that captures various properties scattered in the literature. In the minimization setting, we obtain rates that are sharper than existing ones.}, number={3}, journal={Set-Valued and Variational Analysis}, author={Bui, Minh N. and Combettes, Patrick L.}, year={2021}, month={Sep}, pages={583–603} }
 @article{combettes_pesquet_2021, title={Fixed Point Strategies in Data Science}, volume={69}, DOI={10.1109/TSP.2021.3069677}, abstractNote={The goal of this article is to promote the use of fixed point strategies in data science by showing that they provide a simplifying and unifying framework to model, analyze, and solve a great variety of problems. They are seen to constitute a natural environment to explain the behavior of advanced convex optimization methods as well as of recent nonlinear methods in data science which are formulated in terms of paradigms that go beyond minimization concepts and involve constructs such as Nash equilibria or monotone inclusions. We review the pertinent tools of fixed point theory and describe the main state-of-the-art algorithms for provenly convergent fixed point construction. We also incorporate additional ingredients such as stochasticity, block-implementations, and non-Euclidean metrics, which provide further enhancements. Applications to signal and image processing, machine learning, statistics, neural networks, and inverse problems are discussed.}, journal={IEEE Transactions on Signal Processing}, author={Combettes, Patrick L. and Pesquet, Jean-Christophe}, year={2021}, pages={3878–3905} }
 @article{bui_combettes_2021, title={Multivariate Monotone Inclusions in Saddle Form}, volume={12}, ISSN={["1526-5471"]}, DOI={10.1287/moor.2021.1161}, abstractNote={ We propose a novel approach to monotone operator splitting based on the notion of a saddle operator. Under investigation is a highly structured multivariate monotone inclusion problem involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators, as well as various monotonicity-preserving operations among them. This model encompasses most formulations found in the literature. A limitation of existing primal-dual algorithms is that they operate in a product space that is too small to achieve full splitting of our problem in the sense that each operator is used individually. To circumvent this difficulty, we recast the problem as that of finding a zero of a saddle operator that acts on a bigger space. This leads to an algorithm of unprecedented flexibility, which achieves full splitting, exploits the specific attributes of each operator, is asynchronous, and requires to activate only blocks of operators at each iteration, as opposed to activating all of them. The latter feature is of critical importance in large-scale problems. The weak convergence of the main algorithm is established, as well as the strong convergence of a variant. Various applications are discussed, and instantiations of the proposed framework in the context of variational inequalities and minimization problems are presented. }, journal={MATHEMATICS OF OPERATIONS RESEARCH}, author={Bui, Minh N. and Combettes, Patrick L.}, year={2021}, month={Dec} }
 @article{combettes_woodstock_2021, title={Reconstruction of functions from prescribed proximal points}, volume={268}, DOI={10.1016/j.jat.2021.105606}, abstractNote={Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be represented using firmly nonexpansive operators, even when the original observation process is discontinuous. The proposed framework thus captures a large body of classical and contemporary best approximation problems arising in areas such as harmonic analysis, statistics, interpolation theory, and signal processing. The resulting problem is recast in terms of a common fixed point problem and solved with a new block-iterative algorithm that features approximate projections onto the individual sets as well as an extrapolated relaxation scheme that exploits the possible presence of affine constraints. A numerical application to signal recovery is demonstrated.}, journal={Journal of Approximation Theory}, author={Combettes, Patrick L. and Woodstock, Zev C.}, year={2021}, month={Aug}, pages={105606} }
 @article{combettes_mueller_2021, title={Regression Models for Compositional Data: General Log-Contrast Formulations, Proximal Optimization, and Microbiome Data Applications}, volume={13}, ISBN={1867-1772}, ISSN={1867-1764 1867-1772}, url={http://dx.doi.org/10.1007/s12561-020-09283-2}, DOI={10.1007/s12561-020-09283-2}, abstractNote={Abstract}, number={2}, journal={Statistics in Biosciences}, publisher={Springer Science and Business Media LLC}, author={Combettes, Patrick L. and Mueller, Christian L.}, year={2021}, pages={217–242} }
 @article{combettes_glaudin_2021, title={Solving Composite Fixed Point Problems with Block Updates}, volume={10}, ISBN={2191-950X}, DOI={10.1515/anona-2020-0173}, abstractNote={Abstract}, number={1}, journal={Advances in Nonlinear Analysis}, author={Combettes, Patrick L. and Glaudin, Lilian E.}, year={2021}, pages={1154–1177} }
 @article{combettes_pesquet_2020, title={Deep Neural Network Structures Solving Variational Inequalities}, volume={28}, ISSN={1877-0533 1877-0541}, url={http://dx.doi.org/10.1007/s11228-019-00526-z}, DOI={10.1007/s11228-019-00526-z}, abstractNote={Motivated by structures that appear in deep neural networks, we investigate nonlinear composite models alternating proximity and affine operators defined on different spaces. We first show that a wide range of activation operators used in neural networks are actually proximity operators. We then establish conditions for the averagedness of the proposed composite constructs and investigate their asymptotic properties. It is shown that the limit of the resulting process solves a variational inequality which, in general, does not derive from a minimization problem. The analysis relies on tools from monotone operator theory and sheds some light on a class of neural networks structures with so far elusive asymptotic properties.}, number={3}, journal={Set-Valued and Variational Analysis}, publisher={Springer Science and Business Media LLC}, author={Combettes, Patrick L. and Pesquet, Jean-Christophe}, year={2020}, month={Feb}, pages={491–518} }
 @article{combettes_pesquet_2020, title={Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operators}, volume={2}, ISSN={2577-0187}, url={http://dx.doi.org/10.1137/19m1272780}, DOI={10.1137/19m1272780}, abstractNote={Obtaining sharp Lipschitz constants for feed-forward neural networks is essential to assess their robustness in the face of perturbations of their inputs. We derive such constants in the context of...}, number={2}, journal={SIAM Journal on Mathematics of Data Science}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Combettes, Patrick L. and Pesquet, Jean-Christophe}, year={2020}, month={Jan}, pages={529–557} }
 @article{combettes_müller_2020, title={Perspective maximum likelihood-type estimation via proximal decomposition}, volume={14}, ISSN={1935-7524}, url={http://dx.doi.org/10.1214/19-ejs1662}, DOI={10.1214/19-EJS1662}, abstractNote={We introduce an optimization model for maximum likelihood-type estimation (M-estimation) that generalizes a large class of existing statistical models, including Huber's concomitant M-estimator, Owen's Huber/Berhu concomitant estimator, the scaled lasso, support vector machine regression, and penalized estimation with structured sparsity. The model, termed perspective M-estimation, leverages the observation that convex M-estimators with concomitant scale as well as various regularizers are instances of perspective functions. Such functions are amenable to proximal analysis, which leads to principled and provably convergent optimization algorithms via proximal splitting. Using a geometrical approach based on duality, we derive novel proximity operators for several perspective functions of interest. Numerical experiments on synthetic and real-world data illustrate the broad applicability of the proposed framework.}, number={1}, journal={Electronic Journal of Statistics}, publisher={Institute of Mathematical Statistics}, author={Combettes, Patrick L. and Müller, Christian L.}, year={2020}, pages={207–238} }
 @article{bùi_combettes_2020, title={The Douglas--Rachford Algorithm Converges Only Weakly}, volume={58}, ISSN={0363-0129 1095-7138}, url={http://dx.doi.org/10.1137/19m1308451}, DOI={10.1137/19M1308451}, abstractNote={We show that the weak convergence of the Douglas--Rachford algorithm for finding a zero of the sum of two maximally monotone operators cannot be improved to strong convergence. Likewise, we show that strong convergence can fail for the method of partial inverses.}, number={2}, journal={SIAM Journal on Control and Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Bùi, Minh N. and Combettes, Patrick L.}, year={2020}, month={Jan}, pages={1118–1120} }
 @article{bùi_combettes_2020, title={Warped proximal iterations for monotone inclusions}, volume={491}, ISSN={0022-247X}, url={http://dx.doi.org/10.1016/j.jmaa.2020.124315}, DOI={10.1016/j.jmaa.2020.124315}, abstractNote={Resolvents of set-valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions. We propose a generalization of this notion, called warped resolvent, which is constructed with the help of an auxiliary operator. The properties of warped resolvents are investigated and connections are made with existing notions. Abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems.}, number={1}, journal={Journal of Mathematical Analysis and Applications}, publisher={Elsevier BV}, author={Bùi, Minh N. and Combettes, Patrick L.}, year={2020}, month={Nov}, pages={124315} }
 @inproceedings{combettes_glaudin_2019, place={Hoboken, NJ}, title={Fully Proximal Splitting Algorithms In Image Recovery}, url={http://dx.doi.org/10.23919/eusipco.2019.8903039}, DOI={10.23919/eusipco.2019.8903039}, abstractNote={Structured convex optimization problems in image recovery typically involve a mix of smooth and nonsmooth functions. The common practice is to activate the smooth functions via their gradient and the nonsmooth ones via their proximity operator. We show that, although intuitively natural, this approach is not necessarily the most efficient numerically and that, in particular, activating all the functions proximally may be advantageous. To make this viewpoint viable computationally, we derive a number of new examples of proximity operators of smooth convex functions arising in applications.}, booktitle={2019 27th European Signal Processing Conference (EUSIPCO)}, publisher={IEEE}, author={Combettes, Patrick L. and Glaudin, Lilian E.}, year={2019}, month={Sep} }
 @article{combettes_mcdonald_micchelli_pontil_2019, title={Learning with optimal interpolation norms}, volume={81}, ISBN={1572-9265}, DOI={10.1007/s11075-018-0568-1}, abstractNote={We analyze a class of norms defined via an optimal interpolation problem involving the composition of norms and a linear operator. This construction, known as infimal postcomposition in convex analysis, is shown to encompass various norms which have been used as regularizers in machine learning, signal processing, and statistics. In particular, these include the latent group lasso, the overlapping group lasso, and certain norms used for learning tensors. We establish basic properties of this class of norms and we provide dual norms. The extension to more general classes of convex functions is also discussed. A stochastic block-coordinate version of the Douglas-Rachford algorithm is devised to solve minimization problems involving these regularizers. A prominent feature of the algorithm is that it yields iterates that converge to a solution in the case of nonsmooth losses and random block updates. Finally, we present numerical experiments with problems employing the latent group lasso penalty.}, number={2}, journal={Numerical Algorithms}, author={Combettes, Patrick L. and McDonald, Andrew M. and Micchelli, Charles A. and Pontil, Massimiliano}, year={2019}, month={Jun}, pages={695–717} }
 @article{combettes_glaudin_2019, title={Proximal Activation of Smooth Functions in Splitting Algorithms for Convex Image Recovery}, volume={12}, ISBN={1936-4954}, DOI={10.1137/18M1224763}, abstractNote={Structured convex optimization problems typically involve a mix of smooth and nonsmooth functions. The common practice is to activate the smooth functions via their gradient and the nonsmooth ones via their proximity operator. We show that, although intuitively natural, this approach is not necessarily the most efficient numerically and that, in particular, activating all the functions proximally may be advantageous. To make this viewpoint viable computationally, we derive a number of new examples of proximity operators of smooth convex functions arising in applications. A novel variational model to relax inconsistent convex feasibility problems is also investigated within the proposed framework. Several numerical applications to image recovery are presented to compare the behavior of fully proximal versus mixed proximal/gradient implementations of several splitting algorithms.}, number={4}, journal={SIAM Journal on Imaging Sciences}, author={Combettes, Patrick L. and Glaudin, Lilian E.}, year={2019}, pages={1905–1935} }
 @article{combettes_pesquet_2019, title={Stochastic quasi-Fejer block-coordinate fixed point iterations with random sweeping II: mean-square and linear convergence}, volume={174}, ISBN={1436-4646}, DOI={10.1007/s10107-018-1296-y}, abstractNote={Reference [11] investigated the almost sure weak convergence of block-coordinate fixed point algorithms and discussed their applications to nonlinear analysis and optimization. This algorithmic framework features random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and it allows for stochastic errors in the evaluation of the operators. The present paper establishes results on the mean-square and linear convergence of the iterates. Applications to monotone operator splitting and proximal optimization algorithms are presented.}, number={1-2}, journal={Mathematical Programming}, author={Combettes, Patrick L. and Pesquet, Jean-Christophe}, year={2019}, month={Mar}, pages={433–451} }
 @article{combettes_salzo_villa_2018, title={Consistent learning by composite proximal thresholding}, volume={167}, ISSN={["1436-4646"]}, DOI={10.1007/s10107-017-1133-8}, abstractNote={We investigate the modeling and the numerical solution of machine learning problems with prediction functions which are linear combinations of elements of a possibly infinite dictionary of functions. We propose a novel flexible composite regularization model, which makes it possible to incorporate various priors on the coefficients of the prediction function, including sparsity and hard constraints. We show that the estimators obtained by minimizing the regularized empirical risk are consistent in a statistical sense, and we design an error-tolerant composite proximal thresholding algorithm for computing such estimators. New results on the asymptotic behavior of the proximal forward–backward splitting method are derived and exploited to establish the convergence properties of the proposed algorithm. In particular, our method features a o(1 / m) convergence rate in objective values.}, number={1}, journal={MATHEMATICAL PROGRAMMING}, author={Combettes, Patrick L. and Salzo, Saverio and Villa, Silvia}, year={2018}, month={Jan}, pages={99–127} }
 @inproceedings{combettes_pesquet_2018, title={Linear convergence of stochastic block-coordinate fixed point algorithms}, DOI={10.23919/EUSIPCO.2018.8552941}, abstractNote={Recent random block-coordinate fixed point algorithms are particularly well suited to large-scale optimization in signal and image processing. These algorithms feature random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and they allow for stochastic errors in the evaluation of the operators. The present paper provides new linear convergence results. These convergence rates are compared to those of standard deterministic algorithms both theoretically and experimentally in an image recovery problem.}, booktitle={Proceedings of the European Signal Processing Conference}, author={Combettes, P.L. and Pesquet, J.-C.}, year={2018}, pages={747–751} }
 @article{combettes_2018, title={Monotone operator theory in convex optimization}, volume={170}, ISSN={["1436-4646"]}, DOI={10.1007/s10107-018-1303-3}, abstractNote={Several aspects of the interplay between monotone operator theory and convex optimization are presented. The crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized. We review the properties of subdifferentials as maximally monotone operators and, in tandem, investigate those of proximity operators as resolvents. In particular, we study new transformations which map proximity operators to proximity operators, and establish connections with self-dual classes of firmly nonexpansive operators. In addition, new insights and developments are proposed on the algorithmic front.}, number={1}, journal={MATHEMATICAL PROGRAMMING}, author={Combettes, Patrick L.}, year={2018}, month={Jul}, pages={177–206} }
 @article{combettes_2018, title={Perspective Functions: Properties, Constructions, and Examples}, volume={26}, ISSN={["1877-0541"]}, DOI={10.1007/s11228-017-0407-x}, abstractNote={Many functions encountered in applied mathematics and in statistical data analysis can be expressed in terms of perspective functions. One of the earliest examples is the Fisher information, which appeared in statistics in the 1920s. We analyze various algebraic and convex-analytical properties of perspective functions and provide general schemes to construct lower semicontinuous convex functions from them. Several new examples are presented and existing instances are featured as special cases.}, number={2}, journal={SET-VALUED AND VARIATIONAL ANALYSIS}, author={Combettes, Patrick L.}, year={2018}, month={Jun}, pages={247–264} }
 @article{combettes_muller_2018, title={Perspective functions: Proximal calculus and applications in high-dimensional statistics}, volume={457}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2016.12.021}, abstractNote={Perspective functions arise explicitly or implicitly in various forms in applied mathematics and in statistical data analysis. To date, no systematic strategy is available to solve the associated, typically nonsmooth, optimization problems. In this paper, we fill this gap by showing that proximal methods provide an efficient framework to model and solve problems involving perspective functions. We study the construction of the proximity operator of a perspective function under general assumptions and present important instances in which the proximity operator can be computed explicitly or via straightforward numerical operations. These results constitute central building blocks in the design of proximal optimization algorithms. We showcase the versatility of the framework by designing novel proximal algorithms for state-of-the-art regression and variable selection schemes in high-dimensional statistics.}, number={2}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Combettes, Patrick L. and Muller, Christian L.}, year={2018}, month={Jan}, pages={1283–1306} }
 @article{combettes_salzo_villa_2018, title={Regularized learning schemes in feature Banach spaces}, volume={16}, ISSN={["1793-6861"]}, DOI={10.1142/s0219530516500202}, abstractNote={ This paper proposes a unified framework for the investigation of constrained learning theory in reflexive Banach spaces of features via regularized empirical risk minimization. The focus is placed on Tikhonov-like regularization with totally convex functions. This broad class of regularizers provides a flexible model for various priors on the features, including, in particular, hard constraints and powers of Banach norms. In such context, the main results establish a new general form of the representer theorem and the consistency of the corresponding learning schemes under general conditions on the loss function, the geometry of the feature space, and the modulus of total convexity of the regularizer. In addition, the proposed analysis gives new insight into basic tools such as reproducing Banach spaces, feature maps, and universality. Even when specialized to Hilbert spaces, this framework yields new results that extend the state of the art. }, number={1}, journal={ANALYSIS AND APPLICATIONS}, author={Combettes, Patrick L. and Salzo, Saverio and Villa, Silvia}, year={2018}, month={Jan}, pages={1–54} }
 @article{barlaud_belhajali_combettes_fillatre_2017, title={Classification and Regression Using an Outer Approximation Projection-Gradient Method}, volume={65}, ISSN={["1941-0476"]}, DOI={10.1109/tsp.2017.2709262}, abstractNote={This paper deals with sparse feature selection and grouping for classification and regression. The classification or regression problems under consideration consists of minimizing a convex empirical risk function subject to an <inline-formula><tex-math notation="LaTeX">$\ell ^1$</tex-math></inline-formula> constraint, a pairwise <inline-formula> <tex-math notation="LaTeX">$\ell ^\infty$</tex-math></inline-formula> constraint, or a pairwise <inline-formula> <tex-math notation="LaTeX">$\ell ^1$</tex-math></inline-formula> constraint. Existing work, such as the Lasso formulation, has focused mainly on Lagrangian penalty approximations, which often require <italic>ad hoc</italic> or computationally expensive procedures to determine the penalization parameter. We depart from this approach and address the constrained problem directly via a splitting method. The structure of the method is that of the classical gradient-projection algorithm, which alternates a gradient step on the objective and a projection step onto the lower level set modeling the constraint. The novelty of our approach is that the projection step is implemented via an outer approximation scheme in which the constraint set is approximated by a sequence of simple convex sets consisting of the intersection of two half-spaces. Convergence of the iterates generated by the algorithm is established for a general smooth convex minimization problem with inequality constraints. Experiments on both synthetic and biological data show that our method outperforms penalty methods.}, number={17}, journal={IEEE TRANSACTIONS ON SIGNAL PROCESSING}, author={Barlaud, Michel and Belhajali, Wafa and Combettes, Patrick L. and Fillatre, Lionel}, year={2017}, month={Sep}, pages={4635–4644} }
 @book{bauschke_combettes_2017, place={Cham, Switzerland}, edition={2nd}, series={CMS Books in Mathematics}, title={Convex Analysis and Monotone Operator Theory in Hilbert Spaces}, ISBN={9783319483108 9783319483115}, ISSN={1613-5237 2197-4152}, url={http://dx.doi.org/10.1007/978-3-319-48311-5}, DOI={10.1007/978-3-319-48311-5}, abstractNote={This reference text, now in its second edition, offers a modern unifying presentation of three basic areas of nonlinear analysis: convex analysis, monotone operator theory, and the fixed point theory}, journal={CMS Books in Mathematics}, publisher={Springer International Publishing}, author={Bauschke, Heinz H. and Combettes, Patrick L.}, year={2017}, collection={CMS Books in Mathematics} }
 @article{combettes_glaudin_2017, title={QUASI-NONEXPANSIVE ITERATIONS ON THE AFFINE HULL OF ORBITS: FROM MANN'S MEAN VALUE ALGORITHM TO INERTIAL METHODS}, volume={27}, ISSN={["1095-7189"]}, DOI={10.1137/17m112806x}, abstractNote={Fixed point iterations play a central role in the design and the analysis of a large number of optimization algorithms. We study a new iterative scheme in which the update is obtained by applying a composition of quasi-nonexpansive operators to a point in the affine hull of the orbit generated up to the current iterate. This investigation unifies several algorithmic constructs, including Mann's mean value method, inertial methods, and multilayer memoryless methods. It also provides a framework for the development of new algorithms, such as those we propose for solving monotone inclusion and minimization problems.}, number={4}, journal={SIAM JOURNAL ON OPTIMIZATION}, author={Combettes, Patrick L. and Glaudin, Lilian E.}, year={2017}, pages={2356–2380} }
 @article{combettes_eckstein_2016, title={Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions}, volume={168}, ISSN={0025-5610 1436-4646}, url={http://dx.doi.org/10.1007/s10107-016-1044-0}, DOI={10.1007/s10107-016-1044-0}, abstractNote={We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. The principal innovation in these algorithms is that they are block-iterative in the sense that, at each iteration, only a subset of the monotone operators needs to be processed, as opposed to all operators as in established methods. Flexible strategies are used to select the blocks of operators activated at each iteration. In addition, we allow lags in operator processing, permitting asynchronous implementation. The decomposition phase of each iteration of our methods is to generate points in the graphs of the selected monotone operators, in order to construct a half-space containing the Kuhn–Tucker set associated with the system. The coordination phase of each iteration involves a projection onto this half-space. We present two related methods: the first method provides weakly convergent primal and dual sequences under general conditions, while the second is a variant in which strong convergence is guaranteed without additional assumptions. Neither algorithm requires prior knowledge of bounds on the linear operators involved or the inversion of linear operators. Our algorithmic framework unifies and significantly extends the approaches taken in earlier work on primal-dual projective splitting methods.}, number={1-2}, journal={Mathematical Programming}, publisher={Springer Nature}, author={Combettes, Patrick L. and Eckstein, Jonathan}, year={2016}, month={Jul}, pages={645–672} }
 @article{combettes_nguyen_2016, title={Solving composite monotone inclusions in reflexive Banach spaces by constructing best Bregman approximations from their Kuhn-Tucker set}, volume={23}, number={2}, journal={Journal of Convex Analysis}, author={Combettes, P.L. and Nguyen, Q.V.}, year={2016}, month={May}, pages={481–510} }
 @article{combettes_pesquet_2016, title={Stochastic approximations and perturbations in forward-backward splitting for monotone operators}, volume={1}, url={http://www.yokohamapublishers.jp/online2/oppafa/vol1/p13.html}, number={1}, journal={Pure and Applied Functional Analysis}, author={Combettes, P.L. and Pesquet, J.-C.}, year={2016}, month={Jan}, pages={13–37} }
 @inproceedings{combettes_pesquet_2016, title={Stochastic forward-backward and primal-dual approximation algorithms with application to online image restoration}, DOI={10.1109/EUSIPCO.2016.7760561}, abstractNote={Stochastic approximation techniques have been used in various contexts in data science. We propose a stochastic version of the forward-backward algorithm for minimizing the sum of two convex functions, one of which is not necessarily smooth. Our framework can handle stochastic approximations of the gradient of the smooth function and allows for stochastic errors in the evaluation of the proximity operator of the nonsmooth function. The almost sure convergence of the iterates generated by the algorithm to a minimizer is established under relatively mild assumptions. We also propose a stochastic version of a popular primal-dual proximal splitting algorithm, establish its convergence, and apply it to an online image restoration problem.}, booktitle={Proceedings of the European Signal Processing Conference}, author={Combettes, P.L. and Pesquet, J.-C.}, year={2016} }
 @article{attouch_briceño-arias_combettes_2015, title={A strongly convergent primal–dual method for nonoverlapping domain decomposition}, volume={133}, ISSN={0029-599X 0945-3245}, url={http://dx.doi.org/10.1007/s00211-015-0751-4}, DOI={10.1007/s00211-015-0751-4}, abstractNote={We propose a primal–dual parallel proximal splitting method for solving domain decomposition problems for partial differential equations. The problem is formulated via minimization of energy functions on the subdomains with coupling constraints which model various properties of the solution at the interfaces. The proposed method can handle a wide range of linear and nonlinear problems, with flexible, possibly nonlinear, transmission conditions across the interfaces. Strong convergence in the energy spaces is established in this general setting, and without any additional assumption on the energy functions or the geometry of the problem. Several examples are presented.}, number={3}, journal={Numerische Mathematik}, publisher={Springer Science and Business Media LLC}, author={Attouch, Hédy and Briceño-Arias, Luis M. and Combettes, Patrick L.}, year={2015}, month={Jul}, pages={443–470} }
 @article{alotaibi_combettes_shahzad_2015, title={Best Approximation from the Kuhn-Tucker Set of Composite Monotone Inclusions}, volume={36}, ISSN={0163-0563 1532-2467}, url={http://dx.doi.org/10.1080/01630563.2015.1077864}, DOI={10.1080/01630563.2015.1077864}, abstractNote={Kuhn-Tucker points play a fundamental role in the analysis and the numerical solution of monotone inclusion problems, providing in particular both primal and dual solutions. We propose a class of strongly convergent algorithms for constructing the best approximation to a reference point from the set of Kuhn-Tucker points of a general Hilbertian composite monotone inclusion problem. Applications to systems of coupled monotone inclusions are presented. Our framework does not impose additional assumptions on the operators present in the formulation, and it does not require knowledge of the norm of the linear operators involved in the compositions or the inversion of linear operators.}, number={12}, journal={Numerical Functional Analysis and Optimization}, publisher={Informa UK Limited}, author={Alotaibi, Abdullah and Combettes, Patrick L. and Shahzad, Naseer}, year={2015}, month={Dec}, pages={1513–1532} }
 @article{combettes_yamada_2015, title={Compositions and convex combinations of averaged nonexpansive operators}, volume={425}, ISSN={0022-247X}, url={http://dx.doi.org/10.1016/j.jmaa.2014.11.044}, DOI={10.1016/j.jmaa.2014.11.044}, abstractNote={Properties of compositions and convex combinations of averaged nonexpansive operators are investigated and applied to the design of new fixed point algorithms in Hilbert spaces. An extended version of the forward–backward splitting algorithm for finding a zero of the sum of two monotone operators is obtained.}, number={1}, journal={Journal of Mathematical Analysis and Applications}, publisher={Elsevier BV}, author={Combettes, Patrick L. and Yamada, Isao}, year={2015}, month={May}, pages={55–70} }
 @article{combettes_dũng_2015, title={Kolmogorov n-Widths of Function Classes Induced by a Non-Degenerate Differential Operator: A Convex Duality Approach}, volume={24}, ISSN={1877-0533 1877-0541}, url={http://dx.doi.org/10.1007/s11228-015-0338-3}, DOI={10.1007/s11228-015-0338-3}, abstractNote={The problem of computing the asymptotic order of the Kolmogorov n-width of the unit ball of the space of multivariate periodic functions induced by a differential operator associated with a polynomial in the general case when the ball is compactly embedded into L 2 has been open for a long time. In the present paper, we use convex analytical tools to solve it in the case when the differential operator is non-degenerate.}, number={1}, journal={Set-Valued and Variational Analysis}, publisher={Springer Science and Business Media LLC}, author={Combettes, Patrick L. and Dũng, Dinh}, year={2015}, month={Aug}, pages={83–99} }
 @article{combettes_pesquet_2015, title={Stochastic Quasi-Fejér Block-Coordinate Fixed Point Iterations with Random Sweeping}, volume={25}, ISSN={1052-6234 1095-7189}, url={http://dx.doi.org/10.1137/140971233}, DOI={10.1137/140971233}, abstractNote={This work proposes block-coordinate fixed point algorithms with applications to nonlinear analysis and optimization in Hilbert spaces. The asymptotic analysis relies on a notion of stochastic quasi-Fejer monotonicity, which is thoroughly investigated. The iterative methods under consideration feature random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and they allow for stochastic errors in the evaluation of the operators. Algorithms using quasi-nonexpansive operators or compositions of averaged nonexpansive operators are constructed, and weak and strong convergence results are established for the sequences they generate. As a by-product, novel block-coordinate operator splitting methods are obtained for solving structured monotone inclusion and convex minimization problems. In particular, the proposed framework leads to random block-coordinate versions of the Douglas--Rachford and forward-backward algorithms and of some of their variant...}, number={2}, journal={SIAM Journal on Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Combettes, Patrick L. and Pesquet, Jean-Christophe}, year={2015}, month={Jan}, pages={1221–1248} }
 @inproceedings{combettes_condat_pesquet_vũ_2014, title={A forward-backward view of some primal-dual optimization methods in image recovery}, DOI={10.1109/ICIP.2014.7025841}, abstractNote={A wide array of image recovery problems can be abstracted into the problem of minimizing a sum of composite convex functions in a Hilbert space. To solve such problems, primal-dual proximal approaches have been developed which provide efficient solutions to large-scale optimization problems. The objective of this paper is to show that a number of existing algorithms can be derived from a general form of the forward-backward algorithm applied in a suitable product space. Our approach also allows us to develop useful extensions of existing algorithms by introducing a variable metric. An illustration to image restoration is provided.}, booktitle={Proceedings of the IEEE International Conference on Image Processing}, author={Combettes, P.L. and Condat, L. and Pesquet, J.-C. and Vũ, B.C.}, year={2014}, pages={4141–4145} }
 @article{alghamdi_alotaibi_combettes_shahzad_2014, title={A primal-dual method of partial inverses for composite inclusions}, volume={8}, ISSN={1862-4472 1862-4480}, url={http://dx.doi.org/10.1007/s11590-014-0734-x}, DOI={10.1007/s11590-014-0734-x}, abstractNote={Spingarn's method of partial inverses has found many applications in nonlinear analysis and in optimization. We show that it can be employed to solve composite monotone inclusions in duality, thus opening a new range of applications for the partial inverse formalism. The versatility of the resulting primal-dual splitting algorithm is illustrated through applications to structured monotone inclusions and optimization.}, number={8}, journal={Optimization Letters}, publisher={Springer Science and Business Media LLC}, author={Alghamdi, Maryam A. and Alotaibi, Abdullah and Combettes, Patrick L. and Shahzad, Naseer}, year={2014}, month={Mar}, pages={2271–2284} }
 @article{becker_combettes_2014, title={An algorithm for splitting parallel sums of linearly composed monotone operators, with applications to signal recovery}, volume={15}, number={1}, journal={Journal of Nonlinear and Convex Analysis}, author={Becker, S.R. and Combettes, P.L.}, year={2014}, month={Jan}, pages={137–159} }
 @article{baillon_combettes_cominetti_2014, title={Asymptotic behavior of compositions of under-relaxed nonexpansive operators}, volume={1}, ISSN={2164-6066}, url={http://dx.doi.org/10.3934/jdg.2014.1.331}, DOI={10.3934/jdg.2014.1.331}, abstractNote={In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.}, number={3}, journal={Journal of Dynamics and Games}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Baillon, Jean-Bernard and Combettes, Patrick L. and Cominetti, Roberto}, year={2014}, month={Jul}, pages={331–346} }
 @article{combettes_hiriart-urruty_théra_2014, title={Modern convex analysis}, volume={148}, DOI={10.1007/s10107-014-0815-8}, journal={Mathematical Programming}, author={Combettes, P.L. and Hiriart-Urruty, J.-B. and Théra, M.}, year={2014}, month={Dec}, pages={1–4} }
 @article{alotaibi_combettes_shahzad_2014, title={Solving Coupled Composite Monotone Inclusions by Successive Fejér Approximations of their Kuhn--Tucker Set}, volume={24}, ISSN={1052-6234 1095-7189}, url={http://dx.doi.org/10.1137/130950616}, DOI={10.1137/130950616}, abstractNote={We propose a new class of primal-dual Fejer monotone algorithms for solving systems of composite monotone inclusions. Our construction is inspired by a framework used by Eckstein and Svaiter for the basic problem of finding a zero of the sum of two monotone operators. At each iteration, points in the graph of the monotone operators present in the model are used to construct a half-space containing the Kuhn--Tucker set associated with the system. The primal-dual update is then obtained via a relaxed projection of the current iterate onto this half-space. An important feature that distinguishes the resulting splitting algorithms from existing ones is that they do not require prior knowledge of bounds on the linear operators involved or the inversion of linear operators.}, number={4}, journal={SIAM Journal on Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Alotaibi, Abdullah and Combettes, Patrick L. and Shahzad, Naseer}, year={2014}, month={Jan}, pages={2076–2095} }
 @inbook{briceño-arias_combettes_2013, title={Monotone Operator Methods for Nash Equilibria in Non-potential Games}, ISBN={9781461476207 9781461476214}, ISSN={2194-1009 2194-1017}, url={http://dx.doi.org/10.1007/978-1-4614-7621-4_9}, DOI={10.1007/978-1-4614-7621-4_9}, abstractNote={We observe that a significant class of Nash equilibrium problems in non-potential games can be associated with monotone inclusion problems. We propose splitting techniques to solve such problems and establish their convergence. Applications to generalized Nash equilibria, zero-sum games, and cyclic proximation problems are demonstrated.}, booktitle={Computational and Analytical Mathematics}, publisher={Springer New York}, author={Briceño-Arias, Luis M. and Combettes, Patrick L.}, year={2013}, pages={143–159} }
 @article{combettes_reyes_2013, title={Moreau’s decomposition in Banach spaces}, volume={139}, ISSN={0025-5610 1436-4646}, url={http://dx.doi.org/10.1007/s10107-013-0663-y}, DOI={10.1007/s10107-013-0663-y}, abstractNote={Moreau’s decomposition is a powerful nonlinear hilbertian analysis tool that has been used in various areas of optimization and applied mathematics. In this paper, it is extended to reflexive Banach spaces and in the context of generalized proximity measures. This extension unifies and significantly improves upon existing results.}, number={1-2}, journal={Mathematical Programming}, publisher={Springer Science and Business Media LLC}, author={Combettes, Patrick L. and Reyes, Noli N.}, year={2013}, month={Mar}, pages={103–114} }
 @article{combettes_2013, title={Systems of Structured Monotone Inclusions: Duality, Algorithms, and Applications}, volume={23}, ISSN={1052-6234 1095-7189}, url={http://dx.doi.org/10.1137/130904160}, DOI={10.1137/130904160}, abstractNote={A general primal-dual splitting algorithm for solving systems of structured coupled monotone inclusions in Hilbert spaces is introduced and its asymptotic behavior is analyzed. Each inclusion in the primal system features compositions with linear operators, parallel sums, and Lipschitzian operators. All the operators involved in this structured model are used separately in the proposed algorithm, most steps of which can be executed in parallel. This provides a flexible solution method applicable to a variety of problems beyond the reach of the state-of-the-art. Several applications are discussed to illustrate this point.}, number={4}, journal={SIAM Journal on Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Combettes, Patrick L.}, year={2013}, month={Jan}, pages={2420–2447} }
 @article{combettes_vũ_2013, title={Variable metric quasi-Fejér monotonicity}, volume={78}, ISSN={0362-546X}, url={http://dx.doi.org/10.1016/j.na.2012.09.008}, DOI={10.1016/j.na.2012.09.008}, abstractNote={The notion of quasi-Fejér monotonicity has proven to be an efficient tool to simplify and unify the convergence analysis of various algorithms arising in applied nonlinear analysis. In this paper, we extend this notion in the context of variable metric algorithms, whereby the underlying norm is allowed to vary at each iteration. Applications to convex optimization and inverse problems are demonstrated.}, journal={Nonlinear Analysis: Theory, Methods & Applications}, publisher={Elsevier BV}, author={Combettes, Patrick L. and Vũ, Bằng C.}, year={2013}, month={Feb}, pages={17–31} }
 @article{baillon_combettes_cominetti_2012, title={There is no variational characterization of the cycles in the method of periodic projections}, volume={262}, ISSN={0022-1236}, url={http://dx.doi.org/10.1016/j.jfa.2011.09.002}, DOI={10.1016/j.jfa.2011.09.002}, abstractNote={The method of periodic projections consists in iterating projections onto m closed convex subsets of a Hilbert space according to a periodic sweeping strategy. In the presence of m⩾3 sets, a long-standing question going back to the 1960s is whether the limit cycles obtained by such a process can be characterized as the minimizers of a certain functional. In this paper we answer this question in the negative. Projection algorithms for minimizing smooth convex functions over a product of convex sets are also discussed.}, number={1}, journal={Journal of Functional Analysis}, publisher={Elsevier BV}, author={Baillon, J.-B. and Combettes, P.L. and Cominetti, R.}, year={2012}, month={Jan}, pages={400–408} }
 @article{combettes_vũ_2012, title={Variable metric forward–backward splitting with applications to monotone inclusions in duality}, volume={63}, ISSN={0233-1934 1029-4945}, url={http://dx.doi.org/10.1080/02331934.2012.733883}, DOI={10.1080/02331934.2012.733883}, abstractNote={We propose a variable metric forward–backward splitting algorithm and prove its convergence in real Hilbert spaces. We then use this framework to derive primal-dual splitting algorithms for solving various classes of monotone inclusions in duality. Some of these algorithms are new even when specialized to the fixed metric case. Various applications are discussed.}, number={9}, journal={Optimization}, publisher={Informa UK Limited}, author={Combettes, Patrick L. and Vũ, Băng C.}, year={2012}, month={Dec}, pages={1289–1318} }
 @article{briceño-arias_combettes_2011, title={A Monotone+Skew Splitting Model for Composite Monotone Inclusions in Duality}, volume={21}, ISSN={1052-6234 1095-7189}, url={http://dx.doi.org/10.1137/10081602x}, DOI={10.1137/10081602x}, abstractNote={The principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew-adjoint operator. An algorithmic framework is developed for solving this generic problem in a Hilbert space setting. New primal-dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators, and convergence results are established. These algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration. Comparisons with existing methods are made and applications to composite variational problems are demonstrated.}, number={4}, journal={SIAM Journal on Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Briceño-Arias, Luis M. and Combettes, Patrick L.}, year={2011}, month={Oct}, pages={1230–1250} }
 @book{bauschke_combettes_2011, place={New York}, series={CMS Books in Mathematics}, title={Convex Analysis and Monotone Operator Theory in Hilbert Spaces}, ISBN={9781441994660 9781441994677}, ISSN={1613-5237}, url={http://dx.doi.org/10.1007/978-1-4419-9467-7}, DOI={10.1007/978-1-4419-9467-7}, abstractNote={This book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space. A concise exposition of related constructive fixed point theory is present}, journal={CMS Books in Mathematics}, publisher={Springer}, author={Bauschke, Heinz H. and Combettes, Patrick L.}, year={2011}, collection={CMS Books in Mathematics} }
 @book{bauschke_burachik_combettes_elser_luke_wolkowicz_2011, place={New York}, series={Springer Optimization and Its Applications}, title={Fixed-Point Algorithms for Inverse Problems in Science and Engineering}, ISBN={9781441995681 9781441995698}, ISSN={1931-6828}, url={http://dx.doi.org/10.1007/978-1-4419-9569-8}, DOI={10.1007/978-1-4419-9569-8}, publisher={Springer}, year={2011}, collection={Springer Optimization and Its Applications} }
 @article{censor_chen_combettes_davidi_herman_2011, title={On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints}, volume={51}, ISSN={0926-6003 1573-2894}, url={http://dx.doi.org/10.1007/s10589-011-9401-7}, DOI={10.1007/s10589-011-9401-7}, abstractNote={The effectiveness of projection methods for solving systems of linear inequalities is investigated. It is shown that they often have a computational advantage over alternatives that have been proposed for solving the same problem and that this makes them successful in many real-world applications. This is supported by experimental evidence provided in this paper on problems of various sizes (up to tens of thousands of unknowns satisfying up to hundreds of thousands of constraints) and by a discussion of the demonstrated efficacy of projection methods in numerous scientific publications and commercial patents (dealing with problems that can have over a billion unknowns and a similar number of constraints).}, number={3}, journal={Computational Optimization and Applications}, publisher={Springer Science and Business Media LLC}, author={Censor, Yair and Chen, Wei and Combettes, Patrick L. and Davidi, Ran and Herman, Gabor T.}, year={2011}, month={Mar}, pages={1065–1088} }
 @article{combettes_pesquet_2011, title={Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators}, volume={20}, ISSN={1877-0533 1877-0541}, url={http://dx.doi.org/10.1007/s11228-011-0191-y}, DOI={10.1007/s11228-011-0191-y}, abstractNote={We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued operators are processed individually via their resolvents. In addition, the algorithm is highly parallel in that most of its steps can be executed simultaneously. This work brings together and notably extends various types of structured monotone inclusion problems and their solution methods. The application to convex minimization problems is given special attention.}, number={2}, journal={Set-Valued and Variational Analysis}, publisher={Springer Science and Business Media LLC}, author={Combettes, Patrick L. and Pesquet, Jean-Christophe}, year={2011}, month={Aug}, pages={307–330} }
 @inbook{combettes_pesquet_2011, title={Proximal Splitting Methods in Signal Processing}, ISBN={9781441995681 9781441995698}, ISSN={1931-6828}, url={http://dx.doi.org/10.1007/978-1-4419-9569-8_10}, DOI={10.1007/978-1-4419-9569-8_10}, abstractNote={The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of inverse problems and, especially, in signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.}, booktitle={Springer Optimization and Its Applications}, publisher={Springer New York}, author={Combettes, Patrick L. and Pesquet, Jean-Christophe}, year={2011}, pages={185–212} }
 @article{combettes_dũng_vũ_2011, title={Proximity for sums of composite functions}, volume={380}, ISSN={0022-247X}, url={http://dx.doi.org/10.1016/j.jmaa.2011.02.079}, DOI={10.1016/j.jmaa.2011.02.079}, abstractNote={We propose an algorithm for computing the proximity operator of a sum of composite convex functions in Hilbert spaces and investigate its asymptotic behavior. Applications to best approximation and image recovery are described.}, number={2}, journal={Journal of Mathematical Analysis and Applications}, publisher={Elsevier BV}, author={Combettes, Patrick L. and Dũng, Đinh and Vũ, Bằng Công}, year={2011}, month={Aug}, pages={680–688} }
 @article{attouch_briceño-arias_combettes_2010, title={A Parallel Splitting Method for Coupled Monotone Inclusions}, volume={48}, ISSN={0363-0129 1095-7138}, url={http://dx.doi.org/10.1137/090754297}, DOI={10.1137/090754297}, abstractNote={A parallel splitting method is proposed for solving systems of coupled monotone inclusions in Hilbert spaces, and its convergence is established under the assumption that solutions exist. Unlike existing alternating algorithms, which are limited to two variables and linear coupling, our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions, variational problems, best approximation, and network flows.}, number={5}, journal={SIAM Journal on Control and Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Attouch, Hédy and Briceño-Arias, Luis M. and Combettes, Patrick L.}, year={2010}, month={Jan}, pages={3246–3270} }
 @inproceedings{bolte_combettes_pesquet_2010, title={Alternating proximal algorithm for blind image recovery}, DOI={10.1109/ICIP.2010.5652173}, abstractNote={We consider a variational formulation of blind image recovery problems. A novel iterative proximal algorithm is proposed to solve the associated nonconvex minimization problem. Under suitable assumptions, this algorithm is shown to have better convergence properties than standard alternating minimization techniques. The objective function includes a smooth convex data fidelity term and nonsmooth convex regularization terms modeling prior information on the data and on the unknown linear degradation operator. A novelty of our approach is to bring into play recent nonsmooth analysis results. The pertinence of the proposed method is illustrated in an image restoration example.}, booktitle={Proceedings of the IEEE International Conference on Image Processing}, author={Bolte, J. and Combettes, P.L. and Pesquet, J.-C.}, year={2010}, pages={1673–1676} }
 @article{combettes_dũng_vũ bằng công_2010, title={Dualization of Signal Recovery Problems}, volume={18}, ISSN={1877-0533 1877-0541}, url={http://dx.doi.org/10.1007/s11228-010-0147-7}, DOI={10.1007/s11228-010-0147-7}, abstractNote={In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenable to solution by current methods but they feature Fenchel–Moreau–Rockafellar dual problems that can be solved by forward-backward splitting. The proposed algorithm produces simultaneously a sequence converging weakly to a dual solution, and a sequence converging strongly to the primal solution. Our framework is shown to capture and extend several existing duality-based signal recovery methods and to be applicable to a variety of new problems beyond their scope.}, number={3-4}, journal={Set-Valued and Variational Analysis}, publisher={Springer Science and Business Media LLC}, author={Combettes, Patrick L. and Dũng, Đinh and Vũ Bằng Công}, year={2010}, month={Sep}, pages={373–404} }
 @article{combettes_reyes_2010, title={Functions with prescribed best linear approximations}, volume={162}, ISSN={0021-9045}, url={http://dx.doi.org/10.1016/j.jat.2009.12.007}, DOI={10.1016/j.jat.2009.12.007}, abstractNote={A common problem in applied mathematics is that of finding a function in a Hilbert space with prescribed best approximations from a finite number of closed vector subspaces. In the present paper we study the question of the existence of solutions to such problems. A finite family of subspaces is said to satisfy the Inverse Best Approximation Property (IBAP) if there exists a point that admits any selection of points from these subspaces as best approximations. We provide various characterizations of the IBAP in terms of the geometry of the subspaces. Connections between the IBAP and the linear convergence rate of the periodic projection algorithm for solving the underlying affine feasibility problem are also established. The results are applied to investigate problems in harmonic analysis, integral equations, signal theory, and wavelet frames.}, number={5}, journal={Journal of Approximation Theory}, publisher={Elsevier BV}, author={Combettes, Patrick L. and Reyes, Noli N.}, year={2010}, month={May}, pages={1095–1116} }
 @article{briceño-arias_combettes_pesquet_pustelnik_2010, title={Proximal Algorithms for Multicomponent Image Recovery Problems}, volume={41}, ISSN={0924-9907 1573-7683}, url={http://dx.doi.org/10.1007/s10851-010-0243-1}, DOI={10.1007/s10851-010-0243-1}, abstractNote={In recent years, proximal splitting algorithms have been applied to various monocomponent signal and image recovery problems. In this paper, we address the case of multicomponent problems. We first provide closed form expressions for several important multicomponent proximity operators and then derive extensions of existing proximal algorithms to the multicomponent setting. These results are applied to stereoscopic image recovery, multispectral image denoising, and image decomposition into texture and geometry components.}, number={1-2}, journal={Journal of Mathematical Imaging and Vision}, publisher={Springer Science and Business Media LLC}, author={Briceño-Arias, L. M. and Combettes, P. L. and Pesquet, J.-C. and Pustelnik, N.}, year={2010}, month={Dec}, pages={3–22} }
 @inproceedings{briceño-arias_combettes_pesquet_pustelnik_2010, title={Proximal method for geometry and texture image decomposition}, DOI={10.1109/ICIP.2010.5653670}, abstractNote={We propose a variational method for decomposing an image into a geometry and a texture component. Our model involves the sum of two functions promoting separately properties of each component, and of a coupling function modeling the interaction between the components. None of these functions is required to be differentiable, which significantly broadens the range of decompositions achievable through variational approaches. The convergence of the proposed proximal algorithm is guaranteed under suitable assumptions. Numerical examples are provided that show an application of the algorithm to image decomposition and restoration in the presence of Poisson noise.}, booktitle={Proceedings of the IEEE International Conference on Image Processing}, author={Briceño-Arias, L.M. and Combettes, P.L. and Pesquet, J.-C. and Pustelnik, N.}, year={2010}, pages={2721–2724} }
 @article{bauschke_combettes_2010, title={The Baillon-Haddad theorem revisited}, volume={17}, number={4}, journal={Journal of Convex Analysis}, author={Bauschke, H.H. and Combettes, P.L.}, year={2010}, month={Dec}, pages={781–787} }
 @article{briceño-arias_combettes_2009, title={Convex variational formulation with smooth coupling for multicomponent signal decomposition and recovery}, volume={2}, DOI={10.4208/nmtma.2009.m9009s}, abstractNote={A convex variational formulation is proposed to solve multicomponent signal processing problems in Hilbert spaces. The cost function consists of a separable term, in which each component is modeled through its own potential, and of a coupling term, in which constraints on linear transformations of the components are penalized with smooth functionals. An algorithm with guaranteed weak convergence to a solution to the problem is provided. Various multicomponent signal decomposition and recovery applications are discussed. AMS subject classifications: 94A12, 90C25}, number={4}, journal={Numerical Mathematics: Theory, Methods, and Applications}, author={Briceño-Arias, L.M. and Combettes, P.L.}, year={2009}, month={Nov}, pages={485–508} }
 @article{combettes_2009, title={Iterative construction of the resolvent of a sum of maximal monotone operators}, volume={16}, number={4}, journal={Journal of Convex Analysis}, author={Combettes, P.L.}, year={2009}, month={Dec}, pages={727–748} }
 @inproceedings{combettes_pesquet_2009, title={Split convex minimization algorithm for signal recovery}, DOI={10.1109/ICASSP.2009.4959676}, abstractNote={A broad range of signal recovery problems can be abstracted into the problem of minimizing the sum of several convex functions in a Hilbert space. We propose a proximal decomposition algorithm which, under mild conditions, provides a solution to such a problem. A significant improvement over the methods currently in use in the area of signal recovery is that it is not limited to two nondifferentiable functions. An application to image restoration is demonstrated.}, booktitle={Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing}, author={Combettes, P.L. and Pesquet, J.-C.}, year={2009}, pages={685–688} }
 @inbook{capricelli_combettes_2008, title={A Convex Programming Algorithm for Noisy Discrete Tomography}, ISBN={9780817636142 9780817645434}, url={http://dx.doi.org/10.1007/978-0-8176-4543-4_10}, DOI={10.1007/978-0-8176-4543-4_10}, abstractNote={A convex programming approach to discrete tomographic image reconstruction in noisy environments is proposed. Conventional constraints are mixed with noise-based constraints on the sinogram and a binariness-promoting total variation constraint. The noise-based constraints are modeled as confidence regions that are constructed under a Poisson noise assumption. A convex objective is then minimized over the resulting feasibility set via a parallel block-iterative method. Applications to binary tomographic reconstruction are demonstrated.}, booktitle={Advances in Discrete Tomography and Its Applications}, publisher={Birkhäuser Boston}, author={Capricelli, T.D. and Combettes, P.L.}, year={2008}, month={Jan}, pages={207–226} }
 @article{bauschke_combettes_2008, title={A Dykstra-like algorithm for two monotone operators}, volume={4}, number={3}, journal={Pacific Journal of Optimization}, author={Bauschke, H.H. and Combettes, P.L.}, year={2008}, month={Sep}, pages={383–391} }
 @article{combettes_pesquet_2008, title={A proximal decomposition method for solving convex variational inverse problems}, volume={24}, ISSN={0266-5611 1361-6420}, url={http://dx.doi.org/10.1088/0266-5611/24/6/065014}, DOI={10.1088/0266-5611/24/6/065014}, abstractNote={A broad range of inverse problems can be abstracted into the problem of minimizing the sum of several convex functions in a Hilbert space. We propose a proximal decomposition algorithm for solving this problem with an arbitrary number of nonsmooth functions and establish its weak convergence. The algorithm fully decomposes the problem in that it involves each function individually via its own proximity operator. A significant improvement over the methods currently in use in the area of inverse problems is that it is not limited to two nonsmooth functions. Numerical applications to signal and image processing problems are demonstrated.}, number={6}, journal={Inverse Problems}, publisher={IOP Publishing}, author={Combettes, Patrick L and Pesquet, Jean-Christophe}, year={2008}, month={Nov}, pages={065014} }
 @article{combettes_pesquet_2008, title={Proximal Thresholding Algorithm for Minimization over Orthonormal Bases}, volume={18}, ISSN={1052-6234 1095-7189}, url={http://dx.doi.org/10.1137/060669498}, DOI={10.1137/060669498}, abstractNote={The notion of soft thresholding plays a central role in problems from various areas of applied mathematics, in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties, providing, in particular, several characterizations of such thresholders. We then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems, and we establish the strong convergence of a proximal thresholding algorithm to solve it. Numerical applications to signal recovery are demonstrated.}, number={4}, journal={SIAM Journal on Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Combettes, Patrick L. and Pesquet, Jean-Christophe}, year={2008}, month={Jan}, pages={1351–1376} }
 @article{combettes_hirstoaga_2008, title={Visco-penalization of the sum of two monotone operators}, volume={69}, ISSN={0362-546X}, url={http://dx.doi.org/10.1016/j.na.2007.06.003}, DOI={10.1016/j.na.2007.06.003}, abstractNote={A new type of approximating curve for finding a particular zero of the sum of two maximal monotone operators in a Hilbert space is investigated.This curve consists of the zeros of perturbed problems in which one operator is replaced with its Yosida approximation and a viscosity term is added.As the perturbation vanishes, the curve is shown to converge to the zero of the sum that solves a particular strictly monotone variational inequality.As an offspring of this result, we obtain an approximating curve for finding a particular zero of the sum of several maximal monotone operators.Applications to convex optimization are discussed.}, number={2}, journal={Nonlinear Analysis: Theory, Methods & Applications}, publisher={Elsevier BV}, author={Combettes, Patrick L. and Hirstoaga, Sever A.}, year={2008}, month={Jul}, pages={579–591} }
 @article{combettes_pesquet_2007, title={A Douglas–Rachford Splitting Approach to Nonsmooth Convex Variational Signal Recovery}, volume={1}, ISSN={1932-4553 1941-0484}, url={http://dx.doi.org/10.1109/jstsp.2007.910264}, DOI={10.1109/jstsp.2007.910264}, abstractNote={Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on the Douglas-Rachford algorithm for monotone operator-splitting, is obtained under general conditions. Applications to non-Gaussian image denoising in a tight frame are also demonstrated.}, number={4}, journal={IEEE Journal of Selected Topics in Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, Patrick L. and Pesquet, Jean-Christophe}, year={2007}, month={Dec}, pages={564–574} }
 @article{chaux_combettes_pesquet_wajs_2007, title={A variational formulation for frame-based inverse problems}, volume={23}, ISSN={0266-5611 1361-6420}, url={http://dx.doi.org/10.1088/0266-5611/23/4/008}, DOI={10.1088/0266-5611/23/4/008}, abstractNote={A convex variational framework is proposed for solving inverse problems in Hilbert spaces with a priori information on the representation of the target solution in a frame. The objective function to be minimized consists of a separable term penalizing each frame coefficient individually, and a smooth term modelling the data formation model as well as other constraints. Sparsity-constrained and Bayesian formulations are examined as special cases. A splitting algorithm is presented to solve this problem and its convergence is established in infinite-dimensional spaces under mild conditions on the penalization functions, which need not be differentiable. Numerical simulations demonstrate applications to frame-based image restoration.}, number={4}, journal={Inverse Problems}, publisher={IOP Publishing}, author={Chaux, Caroline and Combettes, Patrick L and Pesquet, Jean-Christophe and Wajs, Valérie R}, year={2007}, month={Jun}, pages={1495–1518} }
 @inproceedings{chaux_combettes_pesquet_wajs_2007, title={Opérateurs proximaux pour la restauration bayésienne de signaux}, url={http://hdl.handle.net/2042/17744}, booktitle={Proceedings of the Twenty First GRETSI Symposium}, author={Chaux, C. and Combettes, P.L. and Pesquet, J.-C. and Wajs, V.R.}, year={2007}, month={Sep}, pages={1277–1280} }
 @inproceedings{combettes_pesquet_2007, title={Sparse signal recovery by iterative proximal thresholding}, booktitle={Proceedings of the European Signal Processing Conference}, author={Combettes, P.L. and Pesquet, J.-C.}, year={2007} }
 @inproceedings{bauschke_combettes_pesquet_2006, title={A decomposition method for nonsmooth convex variational signal recovery}, volume={5}, DOI={10.1109/ICASSP.2006.1661444}, abstractNote={Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) proper lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on an extension of the Douglas-Rachford algorithm for monotone operators splitting, is established under general conditions. Various signal recovery applications are discussed and numerical results are provided}, booktitle={Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing}, author={Bauschke, H.H. and Combettes, P.L. and Pesquet, J.-C.}, year={2006}, pages={989–992} }
 @article{bauschke_combettes_luke_2006, title={A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space}, volume={141}, ISSN={0021-9045}, url={http://dx.doi.org/10.1016/j.jat.2006.01.003}, DOI={10.1016/j.jat.2006.01.003}, abstractNote={A new iterative method for finding the projection onto the intersection of two closed convex sets in a Hilbert space is presented. It is a Haugazeau-like modification of a recently proposed averaged alternating reflections method which produces a strongly convergent sequence.}, number={1}, journal={Journal of Approximation Theory}, publisher={Elsevier BV}, author={Bauschke, Heinz H. and Combettes, Patrick L. and Luke, D. Russell}, year={2006}, month={Jul}, pages={63–69} }
 @article{combettes_hirstoaga_2006, title={Approximating curves for nonexpansive and monotone operators}, volume={13}, number={3-4}, journal={Journal of Convex Analysis}, author={Combettes, P.L. and Hirstoaga, S.A.}, year={2006}, month={Dec}, pages={633–646} }
 @inproceedings{chaux_combettes_pesquet_wajs_2006, title={Iterative image deconvolution using overcomplete representations}, booktitle={Proceedings of the European Signal Processing Conference}, author={Chaux, C. and Combettes, P.L. and Pesquet, J.C. and Wajs, V.R.}, year={2006} }
 @article{bauschke_combettes_noll_2006, title={Joint minimization with alternating Bregman proximity operators}, volume={2}, number={3}, journal={Pacific Journal of Optimization}, author={Bauschke, H.H. and Combettes, P.L. and Noll, D.}, year={2006}, month={Sep}, pages={401–424} }
 @inproceedings{chaux_combettes_pesquet_wajs_2005, title={A forward-backward algorithm for image restoration with sparse representations}, booktitle={Proceedings of the International Conference on Signal Processing with Adaptative Sparse Structured Representations}, author={Chaux, C. and Combettes, P.L. and Pesquet, J.-C. and Wajs, V.R.}, year={2005}, pages={49–52} }
 @inproceedings{bauschke_combettes_luke_2005, title={A new generation of iterative transform algorithms for phase contrast tomography}, volume={4}, DOI={10.1109/ICASSP.2005.1415952}, abstractNote={Improvements in electromagnetic sources, detectors, optical components, and computational imaging have made it possible to achieve three-dimensional atomic-scale resolution using tomographic phase-contrast imaging techniques. These greater capabilities have placed a premium on improving the efficiency and stability of phase retrieval algorithms for recovering the missing phase information in diffraction observations. In some cases, so called direct methods suffice, but, for large macromolecules and nonperiodic structures, one must rely on numerical techniques for reconstructing the missing phase. This is the principal motivation of our work. We report on recent progress in algorithms for iterative phase retrieval. The theory of convex optimisation is used to develop and to gain insight into counterparts for the nonconvex problem of phase retrieval. We propose a relaxation of averaged alternating reflectors and determine the fundamental mathematical properties of the related operator in the convex case. Numerical studies support our theoretical observations and demonstrate the effectiveness of the newer generation of algorithms compared to the current state of the art.}, booktitle={Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing}, author={Bauschke, H.H. and Combettes, P.L. and Luke, D.R.}, year={2005}, pages={89–92} }
 @article{combettes_hirstoaga_2005, title={Equilibrium programming in Hilbert spaces}, volume={6}, number={1}, journal={Journal of Nonlinear and Convex Analysis}, author={Combettes, P.L. and Hirstoaga, S.A.}, year={2005}, month={Apr}, pages={117–136} }
 @inproceedings{combettes_pesquet_2005, title={Estimating first-order finite-difference information in image restoration problems}, ISBN={0780385543}, url={http://dx.doi.org/10.1109/icip.2004.1418755}, DOI={10.1109/icip.2004.1418755}, abstractNote={First-order finite-difference information has been exploited in a variety of image and signal restoration settings. These approaches typically require - implicitly or explicitly - that certain attributes of the finite-difference images be known a priori. We propose a new statistical framework in which such attributes are estimated a posteriori from the observed data under the assumption that the noise is additive and Gaussian. Our analysis can be directly applied to the construction of property sets in set theoretic estimation methods. The proposed framework is illustrated through an application to image denoising.}, booktitle={2004 International Conference on Image Processing, 2004. ICIP '04.}, publisher={IEEE}, author={Combettes, P.L. and Pesquet, J.}, year={2005}, month={Apr} }
 @article{bauschke_combettes_kruk_2005, title={Extrapolation algorithm for affine-convex feasibility problems}, volume={41}, ISSN={1017-1398 1572-9265}, url={http://dx.doi.org/10.1007/s11075-005-9010-6}, DOI={10.1007/s11075-005-9010-6}, abstractNote={The convex feasibility problem under consideration is to find a common point of a countable family of closed affine subspaces and convex sets in a Hilbert space. To solve such problems, we propose a general parallel block-iterative algorithmic framework in which the affine subspaces are exploited to introduce extrapolated over-relaxations. This framework encompasses a wide range of projection, subgradient projection, proximal, and fixed point methods encountered in various branches of applied mathematics. The asymptotic behavior of the method is investigated and numerical experiments are provided to illustrate the benefits of the extrapolations.}, number={3}, journal={Numerical Algorithms}, publisher={Springer Science and Business Media LLC}, author={Bauschke, Heinz H. and Combettes, Patrick L. and Kruk, Serge G.}, year={2005}, month={Dec}, pages={239–274} }
 @article{capricelli_combettes_2005, title={Parallel Block-Iterative Reconstruction Algorithms for Binary Tomography}, volume={20}, ISSN={1571-0653}, url={http://dx.doi.org/10.1016/j.endm.2005.05.068}, DOI={10.1016/j.endm.2005.05.068}, abstractNote={A convex programming approach to binary tomographic image reconstruction in noisy environments is proposed. Conventional constraints are mixed with new constraints on the sinogram. A convex objective is then minimized over the resulting feasibility set via a parallel block-iterative method. The new constraints involve noise-based confidence regions and a binarity-promoting total variation constraint.}, journal={Electronic Notes in Discrete Mathematics}, publisher={Elsevier BV}, author={Capricelli, Thomas D. and Combettes, Patrick L.}, year={2005}, month={Jul}, pages={263–280} }
 @article{combettes_wajs_2005, title={Signal Recovery by Proximal Forward-Backward Splitting}, volume={4}, ISSN={1540-3459 1540-3467}, url={http://dx.doi.org/10.1137/050626090}, DOI={10.1137/050626090}, abstractNote={We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.}, number={4}, journal={Multiscale Modeling & Simulation}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Combettes, Patrick L. and Wajs, Valérie R.}, year={2005}, month={Jan}, pages={1168–1200} }
 @article{bauschke_combettes_reich_2005, title={The asymptotic behavior of the composition of two resolvents}, volume={60}, ISSN={0362-546X}, url={http://dx.doi.org/10.1016/j.na.2004.07.054}, DOI={10.1016/j.na.2004.07.054}, abstractNote={The asymptotic behavior of the composition of two resolvents in a Hilbert space is investigated. Connections are made between the solutions of associated monotone inclusion problems and their dual versions. The applications provided include a study of an alternating minimization procedure and a new proof of von Neumann's classical result on the method of alternating projections.}, number={2}, journal={Nonlinear Analysis: Theory, Methods & Applications}, publisher={Elsevier BV}, author={Bauschke, Heinz H. and Combettes, Patrick L. and Reich, Simeon}, year={2005}, month={Jan}, pages={283–301} }
 @article{bauschke_combettes_reich_2005, title={The asymptotic behavior of the composition of two resolvents}, volume={60}, ISSN={0362-546X}, url={http://dx.doi.org/10.1016/S0362-546X(04)00344-X}, DOI={10.1016/S0362-546X(04)00344-X}, abstractNote={The asymptotic behavior of the composition of two resolvents in a Hilbert space is investigated. Connections are made between the solutions of associated monotone inclusion problems and their dual versions. The applications provided include a study of an alternating minimization procedure and a new proof of von Neumann's classical result on the method of alternating projections.}, number={2}, journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Bauschke, H and Combettes, P and Reich, S}, year={2005}, month={Jan}, pages={283–301} }
 @inproceedings{combettes_wajs_2005, place={Hoboken, New Jersey}, title={Theoretical analysis of some regularized image denoising methods}, ISBN={0780385543}, url={http://dx.doi.org/10.1109/icip.2004.1419462}, DOI={10.1109/icip.2004.1419462}, abstractNote={Regularization techniques have been in use in signal recovery for over four decades. In this paper, we propose a new synthetic approach to the study of regularization methods in image denoising problems based on Moreau's proximity operators. We exploit the remarkable properties enjoyed by these operators to establish in a systematic fashion a variety of properties of regularized denoising problems and to propose new numerical schemes to solve them.}, booktitle={2004 International Conference on Image Processing, 2004. ICIP '04.}, publisher={IEEE}, author={Combettes, P.L. and Wajs, V.R.}, year={2005}, month={Apr} }
 @inproceedings{capricelli_combettes_2005, title={Éclatement des contraintes en reconstruction tomographique}, author={Capricelli, T.D. and Combettes, P.L.}, year={2005}, month={Sep} }
 @inproceedings{combettes_pesquet_2004, place={Hoboken, New Jersey}, title={Constraint construction in convex set theoretic signal recovery via Stein's principle [image denoising example]}, ISBN={0780384849}, url={http://dx.doi.org/10.1109/icassp.2004.1326382}, DOI={10.1109/icassp.2004.1326382}, abstractNote={Convex set theoretic estimation methods have been shown to be effective in numerous signal recovery problems due to their ability to incorporate a wide range of deterministic and probabilistic information in the form of constraints on the solution. To date, probabilistic information has been used exclusively to constrain statistics of the estimation residual to be consistent with known properties of the noise. In this paper, we propose a new technique to construct constraint sets from probabilistic information based on Stein's identity. In this framework, probabilistic attributes of the signal to be recovered are estimated from the data. The proposed approach is applicable to signal formation models involving additive Gaussian noise and it leads to geometrically simple sets that can easily be handled via projection methods. An application to image denoising is demonstrated.}, booktitle={2004 IEEE International Conference on Acoustics, Speech, and Signal Processing}, publisher={IEEE}, author={Combettes, P.L. and Pesquet, J.-C.}, year={2004}, month={Sep} }
 @article{bauschke_combettes_luke_2004, title={Finding best approximation pairs relative to two closed convex sets in Hilbert spaces}, volume={127}, ISSN={0021-9045}, url={http://dx.doi.org/10.1016/j.jat.2004.02.006}, DOI={10.1016/j.jat.2004.02.006}, abstractNote={We consider the problem of finding a best approximation pair, i.e., two points which achieve the minimum distance between two closed convex sets in a Hilbert space. When the sets intersect, the method under consideration, termed AAR for averaged alternating reflections, is a special instance of an algorithm due to Lions and Mercier for finding a zero of the sum of two maximal monotone operators. We investigate systematically the asymptotic behavior of AAR in the general case when the sets do not necessarily intersect and show that the method produces best approximation pairs provided they exist. Finitely many sets are handled in a product space, in which case the AAR method is shown to coincide with a special case of Spingarn's method of partial inverses.}, number={2}, journal={Journal of Approximation Theory}, publisher={Elsevier BV}, author={Bauschke, Heinz H. and Combettes, Patrick L. and Luke, D.Russell}, year={2004}, month={Apr}, pages={178–192} }
 @article{combettes_pesquet_2004, title={Image Restoration Subject to a Total Variation Constraint}, volume={13}, ISSN={1057-7149}, url={http://dx.doi.org/10.1109/tip.2004.832922}, DOI={10.1109/tip.2004.832922}, abstractNote={Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under constraints. In this paper, we propose an alternative formulation in which total variation is used as a constraint in a general convex programming framework. This approach places no limitation on the incorporation of additional constraints in the restoration process and the resulting optimization problem can be solved efficiently via block-iterative methods. Image denoising and deconvolution applications are demonstrated.}, number={9}, journal={IEEE Transactions on Image Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L. and Pesquet, J.-C.}, year={2004}, month={Sep}, pages={1213–1222} }
 @article{combettes_pennanen_2004, title={Proximal Methods for Cohypomonotone Operators}, volume={43}, ISSN={0363-0129 1095-7138}, url={http://dx.doi.org/10.1137/s0363012903427336}, DOI={10.1137/s0363012903427336}, abstractNote={Conditions are given for the viability and the weak convergence of an inexact, relaxed proximal point algorithm for finding a common zero of countably many cohypomonotone operators in a Hilbert space. In turn, new convergence results are obtained for an extended version of the proximal method of multipliers in nonlinear programming.}, number={2}, journal={SIAM Journal on Control and Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Combettes, Patrick L. and Pennanen, Teemu}, year={2004}, month={Jan}, pages={731–742} }
 @article{combettes_2004, title={Solving monotone inclusions via compositions of nonexpansive averaged operators}, volume={53}, ISSN={0233-1934 1029-4945}, url={http://dx.doi.org/10.1080/02331930412331327157}, DOI={10.1080/02331930412331327157}, abstractNote={A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analysis covers proximal methods for common zero problems as well as for various splitting methods for finding a zero of the sum of monotone operators.}, number={5-6}, journal={Optimization}, publisher={Informa UK Limited}, author={Combettes, Patrick L.}, year={2004}, month={Oct}, pages={475–504} }
 @inproceedings{combettes_pesquet_2004, place={Hoboken, New Jersey}, title={Total variation information in image recovery}, ISBN={0780377508}, url={http://dx.doi.org/10.1109/icip.2003.1247259}, DOI={10.1109/icip.2003.1247259}, abstractNote={Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under a single constraint. In this paper, we propose an alternative framework in which total variation is used as a constraint in a general quadratic programming context. The advantage of this approach is that it allows for a wider range of constraints to be easily incorporated in the recovery process.}, booktitle={Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429)}, publisher={IEEE}, author={Combettes, P.L. and Pesquet, J.C.}, year={2004}, month={Jun} }
 @article{combettes_pesquet_2004, title={Wavelet-constrained image restoration}, volume={2}, ISSN={0219-6913 1793-690X}, url={http://dx.doi.org/10.1142/s0219691304000688}, DOI={10.1142/s0219691304000688}, abstractNote={Image restoration problems can naturally be cast as constrained convex programming problems in which the constraints arise from a priori information and the observation of signals physically related to the image to be recovered. In this paper, the focus is placed on the construction of constraints based on wavelet representations. Using a mix of statistical and convex-analytical tools, we propose a general framework to construct wavelet-based constraints. The resulting optimization problem is then solved with a block-iterative parallel algorithm which offers great flexibility in terms of implementation. Numerical results illustrate an application of the proposed framework.}, number={4}, journal={International Journal of Wavelets, Multiresolution and Information Processing}, publisher={World Scientific Pub Co Pte Lt}, author={Combettes, Patrick L. and Pesquet, Jean Christophe}, year={2004}, month={Dec}, pages={371–389} }
 @article{combettes_2003, title={A block-iterative surrogate constraint splitting method for quadratic signal recovery}, volume={51}, ISSN={1053-587X}, url={http://dx.doi.org/10.1109/tsp.2003.812846}, DOI={10.1109/tsp.2003.812846}, abstractNote={A block-iterative parallel decomposition method is proposed to solve general quadratic signal recovery problems under convex constraints. The proposed method proceeds by local linearizations of blocks of constraints, and it is therefore not sensitive to their analytical complexity. In addition, it naturally lends itself to implementation on parallel computing architectures due to its flexible block-iterative structure. Comparisons with existing methods are carried out, and the case of inconsistent constraints is also discussed. Numerical results are presented.}, number={7}, journal={IEEE Transactions on Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L.}, year={2003}, month={Jul}, pages={1771–1782} }
 @article{bauschke_borwein_combettes_2003, title={Bregman Monotone Optimization Algorithms}, volume={42}, ISSN={0363-0129 1095-7138}, url={http://dx.doi.org/10.1137/s0363012902407120}, DOI={10.1137/s0363012902407120}, abstractNote={A broad class of optimization algorithms based on Bregman distances in Banach spaces is unified around the notion of Bregman monotonicity. A systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block-iterative surrogate Bregman projection schemes. Another key contribution is the introduction of a class of operators that is shown to be intrinsically tied to the notion of Bregman monotonicity and to include the operators commonly found in Bregman optimization methods. Special emphasis is placed on the viability of the algorithms and the importance of Legendre functions in this regard. Various applications are discussed.}, number={2}, journal={SIAM Journal on Control and Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Bauschke, Heinz H. and Borwein, Jonathan M. and Combettes, Patrick L.}, year={2003}, month={Jan}, pages={596–636} }
 @article{bauschke_combettes_2003, title={Construction of best Bregman approximations in reflexive Banach spaces}, volume={131}, ISSN={0002-9939 1088-6826}, url={http://dx.doi.org/10.1090/s0002-9939-03-07050-3}, DOI={10.1090/s0002-9939-03-07050-3}, abstractNote={An iterative method is proposed to construct the Bregman projection of a point onto a countable intersection of closed convex sets in a reflexive Banach space.}, number={12}, journal={Proceedings of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Bauschke, Heinz H. and Combettes, Patrick L.}, year={2003}, month={Dec}, pages={3757–3766} }
 @article{bauschke_combettes_luke_2003, title={Hybrid projection–reflection method for phase retrieval}, volume={20}, ISSN={1084-7529 1520-8532}, url={http://dx.doi.org/10.1364/josaa.20.001025}, DOI={10.1364/josaa.20.001025}, abstractNote={The phase-retrieval problem, fundamental in applied physics and engineering, addresses the question of how to determine the phase of a complex-valued function from modulus data and additional a priori information. Recently we identified two important methods for phase retrieval, namely, Fienup's basic input-output and hybrid input-output (HIO) algorithms, with classical convex projection methods and suggested that further connections between convex optimization and phase retrieval should be explored. Following up on this work, we introduce a new projection-based method, termed the hybrid projection-reflection (HPR) algorithm, for solving phase-retrieval problems featuring nonnegativity constraints in the object domain. Motivated by properties of the HPR algorithm for convex constraints, we recommend an error measure studied by Fienup more than 20 years ago. This error measure, which has received little attention in the literature, lends itself to an easily implementable stopping criterion. In numerical experiments we found the HPR algorithm to be a competitive alternative to the HIO algorithm and the stopping criterion to be reliable and robust.}, number={6}, journal={Journal of the Optical Society of America A}, publisher={The Optical Society}, author={Bauschke, Heinz H. and Combettes, Patrick L. and Luke, D. Russell}, year={2003}, month={Jun}, pages={1025} }
 @inproceedings{combettes_pesquet_2003, place={Hoboken, New Jersey}, title={Image deconvolution with total variation bounds}, ISBN={0780379462}, url={http://dx.doi.org/10.1109/isspa.2003.1224735}, DOI={10.1109/isspa.2003.1224735}, abstractNote={Total variation has been used exclusively as an objective in the formulation of image deconvolution problems. In this paper, we propose an alternative framework in which total variation is used as a constraint. In contrast with the standard approach, this framework requires an a priori bound on the total variation of the original image, while no a priori information on the noise is necessary. Furthermore, it places no limitation on the incorporation of additional constraints in the recovery process and can be solved efficiently via powerful block-iterative methods.}, booktitle={Seventh International Symposium on Signal Processing and Its Applications, 2003. Proceedings.}, publisher={IEEE}, author={Combettes, P.L. and Pesquet, J.-C.}, year={2003} }
 @article{bauschke_combettes_2003, title={Iterating Bregman Retractions}, volume={13}, ISSN={1052-6234 1095-7189}, url={http://dx.doi.org/10.1137/s1052623402410557}, DOI={10.1137/s1052623402410557}, abstractNote={The notion of a Bregman retraction of a closed convex set in Euclidean space is introduced. Bregman retractions include backward Bregman projections and forward Bregman projections, as well as their convex combinations, and are thus quite flexible. The main result on iterating Bregman retractions unifies several convergence results on projection methods for solving convex feasibility problems. It is also used to construct new sequential and parallel algorithms.}, number={4}, journal={SIAM Journal on Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Bauschke, Heinz H. and Combettes, Patrick L.}, year={2003}, month={Jan}, pages={1159–1173} }
 @inproceedings{bauschke_combettes_luke_2003, title={On the structure of some phase retrieval algorithms}, ISBN={0780376226}, url={http://dx.doi.org/10.1109/icip.2002.1040082}, DOI={10.1109/icip.2002.1040082}, abstractNote={The state of the art for solving the phase retrieval problem in two dimensions relies heavily on the algorithms proposed by Gerchbercy, Saxton, and Fienup. Despite the widespread use of these algorithms, current mathematical theory cannot explain their remarkable success. It is already known that the Gerchberg-Saxton algorithm is a nonconvex version of method of alternating projections. In this paper, we show that two other prominent phase retrieval methods also have well known counterparts in the world of convex optimization algorithms: Fienup's basic input-output algorithm corresponds to Dykstra's algorithm, and Fienup's hybrid input-output algorithm can be viewed as an instance of the Douglas-Rachford algorithm. This work provides a theoretical framework to better understand and, potentially, improve existing phase recovery algorithms.}, booktitle={Proceedings. International Conference on Image Processing}, publisher={IEEE}, author={Bauschke, H.H. and Combettes, P.L. and Luke, D.R.}, year={2003}, month={Jun} }
 @inproceedings{combettes_2002, place={Hoboken, New Jersey}, title={A block-iterative quadratic signal recovery algorithm}, ISBN={0780344286}, url={http://dx.doi.org/10.1109/icassp.1998.678136}, DOI={10.1109/icassp.1998.678136}, abstractNote={We propose a block-iterative parallel decomposition method to solve quadratic signal recovery problems under convex constraints. The idea, of the method is to disintegrate the original multi-constraint problem into a sequence of simple quadratic minimizations over the intersection of two half-spaces constructed by linearizing blocks of constraints. The implementation of the algorithm is quite flexible thanks to its block-parallel structure. In addition a wide range of complex constraints can be incorporated since the method does not require exact constraint enforcement at each step but merely approximate enforcement via linearization. An application to deconvolution is demonstrated.}, booktitle={Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181)}, publisher={IEEE}, author={Combettes, P.L.}, year={2002}, month={Nov} }
 @inproceedings{luo_combettes_2002, place={Hoboken, New Jersey}, title={A level-set subgradient projection algorithm for non-differentiable signal restoration with multiple constraints}, ISBN={0780362934}, url={http://dx.doi.org/10.1109/icassp.2000.861924}, DOI={10.1109/icassp.2000.861924}, abstractNote={An new adaptive level-set subgradient projection algorithm is proposed to solve non-differentiable signal recovery problems with multiple convex constraints. The algorithm is described, its convergence is established, and its implementation is discussed. Applications to constrained total variation signal restoration and denoising are demonstrated.}, booktitle={2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100)}, publisher={IEEE}, author={Luo, Jian and Combettes, P.L.}, year={2002}, month={Nov} }
 @inproceedings{combettes_2002, place={Hoboken, New Jersey}, title={A parallel constraint disintegration and approximation scheme for quadratic signal recovery}, ISBN={0780362934}, url={http://dx.doi.org/10.1109/icassp.2000.861901}, DOI={10.1109/icassp.2000.861901}, abstractNote={A block-iterative parallel decomposition method is proposed to solve general quadratic signal recovery problems under convex constraints. Unlike existing schemes, the proposed method proceeds by local linearizations of blocks of constraints and it is therefore not sensitive to their analytical complexity. Implementation-related issues are discussed and an application to signal deconvolution is demonstrated.}, booktitle={2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100)}, publisher={IEEE}, author={Combettes, P.L.}, year={2002}, month={Nov} }
 @article{combettes_luo_2002, title={An adaptive level set method for nondifferentiable constrained image recovery}, volume={11}, ISSN={1057-7149}, url={http://dx.doi.org/10.1109/tip.2002.804527}, DOI={10.1109/tip.2002.804527}, abstractNote={The formulation of a wide variety of image recovery problems leads to the minimization of a convex objective over a convex set representing the constraints derived from a priori knowledge and consistency with the observed signals. In previous years, nondifferentiable objectives have become popular due in part to their ability to capture certain features such as sharp edges. They also arise naturally in minimax inconsistent set theoretic recovery problems. At the same time, the issue of developing reliable numerical algorithms to solve such convex programs in the context of image recovery applications has received little attention. We address this issue and propose an adaptive level set method for nondifferentiable constrained image recovery. The asymptotic properties of the method are analyzed and its implementation is discussed. Numerical experiments illustrate applications to total variation and minimax set theoretic image restoration and denoising problems.}, number={11}, journal={IEEE Transactions on Image Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L. and Luo, Jian}, year={2002}, month={Nov}, pages={1295–1304} }
 @inproceedings{combettes_pesquet_2002, place={Hoboken, New Jersey}, title={Convex multiresolution analysis}, ISBN={0780335120}, url={http://dx.doi.org/10.1109/tfsa.1996.547473}, DOI={10.1109/tfsa.1996.547473}, abstractNote={A standard wavelet multiresolution analysis can be defined via a sequence of projection operators onto a monotone sequence of closed vector subspaces possessing suitable invariance properties. We propose an extension of this framework in which the linear projection operators are replaced by nonlinear retractions onto convex sets. These retractions are chosen so as to provide a recursive, monotone signal approximation scheme. Numerical simulations are also provided.}, booktitle={Proceedings of Third International Symposium on Time-Frequency and Time-Scale Analysis (TFTS-96)}, publisher={IEEE}, author={Combettes, P.L. and Pesquet, J.-C.}, year={2002}, month={Dec} }
 @inproceedings{combettes_2002, place={Hoboken, New Jersey}, title={Convex set theoretic image recovery with inexact projection algorithms}, ISBN={0780367251}, url={http://dx.doi.org/10.1109/icip.2001.959002}, DOI={10.1109/icip.2001.959002}, abstractNote={In image recovery, convex projection methods have been in use for almost two decades. However, while it is well known that projections can seldom be computed exactly, the effect of inexact projections on the behavior of such methods has not yet been investigated. We propose such an analysis and establish conditions on the projection errors under which the theoretical convergence properties of various algorithms remain valid. Our analysis covers sequential, parallel, and block-iterative (subgradient) projection methods for consistent and inconsistent set theoretic image recovery problems. It is shown in particular that parallel projection methods are more robust to errors than sequential methods such as the popular POCS (projection on to convex sets) algorithm.}, booktitle={Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205)}, publisher={IEEE}, author={Combettes, P.L.}, year={2002}, month={Nov} }
 @article{combettes_pennanen_2002, title={Generalized Mann iterates for constructing fixed points in Hilbert spaces}, volume={275}, ISSN={0022-247X}, url={http://dx.doi.org/10.1016/s0022-247x(02)00221-4}, DOI={10.1016/s0022-247x(02)00221-4}, abstractNote={The mean iteration scheme originally proposed by Mann is extended to a broad class of relaxed, inexact fixed point algorithms in Hilbert spaces. Weak and strong convergence results are established under general conditions on the underlying averaging process and the type of operators involved. This analysis significantly widens the range of applications of mean iteration methods. Several examples are given.}, number={2}, journal={Journal of Mathematical Analysis and Applications}, publisher={Elsevier BV}, author={Combettes, Patrick L. and Pennanen, Teemu}, year={2002}, month={Nov}, pages={521–536} }
 @inproceedings{combettes_2002, place={Hoboken, New Jersey}, title={Generalized convex set theoretic image recovery}, ISBN={0780332598}, url={http://dx.doi.org/10.1109/icip.1996.560883}, DOI={10.1109/icip.1996.560883}, abstractNote={In set theoretic image recovery, the constraints which do not yield convex sets in the chosen Hilbert solution space cannot be enforced. In some cases, however, such constraints may yield convex sets in other Hilbert spaces. We introduce a generalized product space formalism, through which constraints that are convex in different Hilbert spaces can be combined. A nonconvex problem with several sets is reduced to a convex problem with two sets in the product space, where it is solved via an alternating projection method. Applications are discussed.}, booktitle={Proceedings of 3rd IEEE International Conference on Image Processing}, publisher={IEEE}, author={Combettes, P.L.}, year={2002}, month={Dec} }
 @inproceedings{combettes_bondon_2002, place={Hoboken, New Jersey}, title={Hard-constrained signal feasibility problems}, ISBN={0818679190}, url={http://dx.doi.org/10.1109/icassp.1997.595313}, DOI={10.1109/icassp.1997.595313}, abstractNote={We consider the problem of synthesizing feasible signals in the presence of inconsistent convex constraints, some of which are hard in the sense that they must absolutely be satisfied. This problem is formalized as that of minimizing an objective function measuring the degree of unfeasibility with respect to the soft constraints over the intersection of the sets associated with the hard constraints. We first investigate the process of aggregating soft constraints in order to define relevant objectives and then address the question of solving the resulting convex programs. Finally, we provide numerical results to illustrate the benefits of our analysis.}, booktitle={1997 IEEE International Conference on Acoustics, Speech, and Signal Processing}, publisher={IEEE Comput. Soc. Press}, author={Combettes, P.L. and Bondon, P.}, year={2002}, month={Nov} }
 @inproceedings{combettes_pesquet_2002, place={Hoboken, New Jersey}, title={Nonlinear multiresolution image analysis via convex projections}, volume={2}, ISBN={0818688211}, url={http://dx.doi.org/10.1109/icip.1998.723646}, DOI={10.1109/icip.1998.723646}, abstractNote={A standard wavelet multiresolution analysis can be defined via a sequence of projectors onto a monotone sequence of closed vector subspaces possessing certain properties. We propose a nonlinear extension of this framework in which the vector subspaces are replaced by convex subsets. These sets are chosen so as to provide a recursive, monotone approximation scheme that allows for various image features to be investigated. Several classes of convex multiresolution analyzes are discussed and numerical applications to image analysis are demonstrated.}, booktitle={Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269)}, publisher={IEEE Comput. Soc}, author={Combettes, P.L. and Pesquet, J.-C.}, year={2002}, month={Nov}, pages={762–765} }
 @inproceedings{puh_combettes_2002, place={Hoboken, New Jersey}, title={Operator theoretic image coding}, ISBN={0780331923}, url={http://dx.doi.org/10.1109/icassp.1996.544812}, DOI={10.1109/icassp.1996.544812}, abstractNote={A general formalism for iterative image coding is introduced. In this framework, each feature of the image to be coded is associated with a nonexpansive operator which leaves invariant all images possessing the feature in question. The decoding scheme consists of finding a common fixed point of these operators, which can be achieved via efficient parallel algorithms. The proposed formalism provides a flexible treatment of the image coding problem and contains as special cases transform coding, fractal coding, set theoretic coding, and vector quantization.}, booktitle={1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Conference Proceedings}, publisher={IEEE}, author={Puh, H. and Combettes, P.L.}, year={2002}, month={Dec} }
 @article{bauschke_combettes_luke_2002, title={Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization}, volume={19}, ISSN={1084-7529 1520-8532}, url={http://dx.doi.org/10.1364/josaa.19.001334}, DOI={10.1364/josaa.19.001334}, abstractNote={The phase retrieval problem is of paramount importance in various areas of applied physics and engineering. The state of the art for solving this problem in two dimensions relies heavily on the pioneering work of Gerchberg, Saxton, and Fienup. Despite the widespread use of the algorithms proposed by these three researchers, current mathematical theory cannot explain their remarkable success. Nevertheless, great insight can be gained into the behavior, the shortcomings, and the performance of these algorithms from their possible counterparts in convex optimization theory. An important step in this direction was made two decades ago when the error reduction algorithm was identified as a nonconvex alternating projection algorithm. Our purpose is to formulate the phase retrieval problem with mathematical care and to establish new connections between well-established numerical phase retrieval schemes and classical convex optimization methods. Specifically, it is shown that Fienup's basic input-output algorithm corresponds to Dykstra's algorithm and that Fienup's hybrid input-output algorithm can be viewed as an instance of the Douglas-Rachford algorithm. We provide a theoretical framework to better understand and, potentially, to improve existing phase recovery algorithms.}, number={7}, journal={Journal of the Optical Society of America A}, publisher={The Optical Society}, author={Bauschke, Heinz H. and Combettes, Patrick L. and Luke, D. Russell}, year={2002}, month={Jul}, pages={1334} }
 @article{bauschke_combettes_2001, title={A Weak-to-Strong Convergence Principle for Fejér-Monotone Methods in Hilbert Spaces}, volume={26}, ISSN={0364-765X 1526-5471}, url={http://dx.doi.org/10.1287/moor.26.2.248.10558}, DOI={10.1287/moor.26.2.248.10558}, abstractNote={ We consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods that makes them strongly convergent without additional assumptions. Several applications are discussed. }, number={2}, journal={Mathematics of Operations Research}, publisher={Institute for Operations Research and the Management Sciences (INFORMS)}, author={Bauschke, Heinz H. and Combettes, Patrick L.}, year={2001}, month={May}, pages={248–264} }
 @inproceedings{combettes_2001, title={Convexité et signal}, booktitle={Actes du Congrès de Mathématiques Appliquées et Industrielles SMAI'01}, author={Combettes, P.L.}, year={2001}, month={May}, pages={6–16} }
 @article{bauschke_borwein_combettes_2001, title={Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces}, volume={3}, ISSN={0219-1997 1793-6683}, url={http://dx.doi.org/10.1142/s0219199701000524}, DOI={10.1142/s0219199701000524}, abstractNote={ The classical notions of essential smoothness, essential strict convexity, and Legendreness for convex functions are extended from Euclidean to Banach spaces. A pertinent duality theory is developed and several useful characterizations are given. The proofs rely on new results on the more subtle behavior of subdifferentials and directional derivatives at boundary points of the domain. In weak Asplund spaces, a new formula allows the recovery of the subdifferential from nearby gradients. Finally, it is shown that every Legendre function on a reflexive Banach space is zone consistent, a fundamental property in the analysis of optimization algorithms based on Bregman distances. Numerous illustrating examples are provided. }, number={4}, journal={Communications in Contemporary Mathematics}, publisher={World Scientific Pub Co Pte Lt}, author={Bauschke, Heinz H. and Borwein, Jonathan M. and Combettes, Patrick L.}, year={2001}, month={Nov}, pages={615–647} }
 @inbook{combettes_2001, place={New York}, title={Fejér-monotonicity in convex optimization}, volume={2}, booktitle={Encyclopedia of Optimization}, publisher={Springer-Verlag}, author={Combettes, P.L.}, editor={Floudas, C.A. and Pardalos, P.M.Editors}, year={2001}, pages={106–114} }
 @article{combettes_2001, title={On the numerical robustness of the parallel projection method in signal synthesis}, volume={8}, ISSN={1070-9908 1558-2361}, url={http://dx.doi.org/10.1109/97.895371}, DOI={10.1109/97.895371}, abstractNote={The parallel projection method (PPM) uses successive averages of projections onto constraint sets to construct a signal that least violates these constraints in an average squared-distance sense. In this paper, we study the robustness of PPM to errors in the computation of the projections. It is shown that the convergence properties of PPM remain valid under a simple summability condition on the relaxed averages of the errors.}, number={2}, journal={IEEE Signal Processing Letters}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L.}, year={2001}, month={Feb}, pages={45–47} }
 @inbook{combettes_2001, title={Quasi-Fejérian Analysis of Some Optimization Algorithms}, ISBN={9780444505958}, ISSN={1570-579X}, url={http://dx.doi.org/10.1016/s1570-579x(01)80010-0}, DOI={10.1016/s1570-579x(01)80010-0}, abstractNote={A quasi-Fejér sequence is a sequence which satisfies the standard Fejér monotonicityproperty to within an additional error term. This notion is studied in detail in a Hilbert space setting and shown to provide a powerful framework to analyze the convergence of a wide range of optimization algorithms in a systematic fashion. A number of convergence theorems covering and extending existing results are thus established. Special emphasis is placed on the design and the analysis of parallel algorithms.}, booktitle={Studies in Computational Mathematics}, publisher={Elsevier}, author={Combettes, Patrick L.}, year={2001}, pages={115–152} }
 @article{combettes_2000, title={Strong Convergence of Block-Iterative Outer Approximation Methods for Convex Optimization}, volume={38}, ISSN={0363-0129 1095-7138}, url={http://dx.doi.org/10.1137/s036301299732626x}, DOI={10.1137/s036301299732626x}, abstractNote={The strong convergence of a broad class of outer approximation methods for minimizing a convex function over the intersection of an arbitrary number of convex sets in a reflexive Banach space is studied in a unified framework. The generic outer approximation algorithm under investigation proceeds by successive minimizations over the intersection of convex supersets of the feasibility set determined in terms of the current iterate and variable blocks of constraints. The convergence analysis involves flexible constraint approximation and aggregation techniques as well as relatively mild assumptions on the constituents of the problem. Various well-known schemes are recovered as special realizations of the generic algorithm and parallel block-iterative extensions of these schemes are devised within the proposed framework. The case of inconsistent constraints is also considered.}, number={2}, journal={SIAM Journal on Control and Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Combettes, Patrick L.}, year={2000}, month={Jan}, pages={538–565} }
 @inproceedings{luo_combettes_1999, title={A subgradient projection algorithm for nondifferentiable signal recovery}, booktitle={Proceedings of the IEEE Workshop on Nonlinear Signal and Image Processing}, author={Luo, J. and Combettes, P.L.}, year={1999}, pages={452–456} }
 @article{combettes_bondon_1999, title={Hard-constrained inconsistent signal feasibility problems}, volume={47}, ISSN={1053-587X}, url={http://dx.doi.org/10.1109/78.782189}, DOI={10.1109/78.782189}, abstractNote={We consider the problem of synthesizing feasible signals in a Hilbert space in the presence of inconsistent convex constraints, some of which must imperatively be satisfied. This problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones. The resulting convex optimization problem is analyzed, and numerical solution schemes are presented along with convergence results. The proposed formalism and its algorithmic framework unify and extend existing approaches to inconsistent signal feasibility problems. An application to signal synthesis is demonstrated.}, number={9}, journal={IEEE Transactions on Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L. and Bondon, P.}, year={1999}, pages={2460–2468} }
 @inproceedings{combettes_bondon_1998, title={Constrained pulse shape synthesis for digital communications}, booktitle={Proceedings of the European Signal Processing Conference}, author={Combettes, P.L. and Bondon, P.}, year={1998}, pages={573–576} }
 @article{combettes_pesquet_1998, title={Convex multiresolution analysis}, volume={20}, ISSN={0162-8828}, url={http://dx.doi.org/10.1109/34.735804}, DOI={10.1109/34.735804}, abstractNote={A standard wavelet multiresolution analysis can be defined via a sequence of projectors onto a monotone sequence of closed vector subspaces possessing certain properties. We propose a nonlinear extension of this framework in which the vector subspaces are replaced by convex subsets. These sets are chosen so as to provide a recursive, monotone approximation scheme that allows for various signal and image features to be investigated. Several classes of convex multiresolution analyses are discussed and numerical applications to signal and image-processing problems are demonstrated.}, number={12}, journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L. and Pesquet, J.-C.}, year={1998}, pages={1308–1318} }
 @article{combettes_1997, title={Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections}, volume={6}, ISSN={1057-7149}, url={http://dx.doi.org/10.1109/83.563316}, DOI={10.1109/83.563316}, abstractNote={Solving a convex set theoretic image recovery problem amounts to finding a point in the intersection of closed and convex sets in a Hilbert space. The projection onto convex sets (POCS) algorithm, in which an initial estimate is sequentially projected onto the individual sets according to a periodic schedule, has been the most prevalent tool to solve such problems. Nonetheless, POCS has several shortcomings: it converges slowly, it is ill suited for implementation on parallel processors, and it requires the computation of exact projections at each iteration. We propose a general parallel projection method (EMOPSP) that overcomes these shortcomings. At each iteration of EMOPSP, a convex combination of subgradient projections onto some of the sets is formed and the update is obtained via relaxation. The relaxation parameter may vary over an iteration-dependent, extrapolated range that extends beyond the interval [0,2] used in conventional projection methods. EMOPSP not only generalizes existing projection-based schemes, but it also converges very efficiently thanks to its extrapolated relaxations. Theoretical convergence results are presented as well as numerical simulations.}, number={4}, journal={IEEE Transactions on Image Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L.}, year={1997}, month={Apr}, pages={493–506} }
 @article{combettes_1997, title={Hilbertian convex feasibility problem: Convergence of projection methods}, volume={35}, ISSN={0095-4616 1432-0606}, url={http://dx.doi.org/10.1007/bf02683333}, DOI={10.1007/bf02683333}, number={3}, journal={Applied Mathematics & Optimization}, publisher={Springer Science and Business Media LLC}, author={Combettes, P. L.}, year={1997}, month={May}, pages={311–330} }
 @inproceedings{combettes_1996, title={Bounded-error models in inverse problems}, volume={2}, booktitle={Proceedings of the 1996 IMACS/IEEE MultiConference on Computational Engineering in Systems Applications}, author={Combettes, P.L.}, year={1996}, pages={1023–1027} }
 @article{combettes_chaussalet_1996, title={Combining statistical information in set theoretic estimation}, volume={3}, ISSN={1070-9908 1558-2361}, url={http://dx.doi.org/10.1109/97.481155}, DOI={10.1109/97.481155}, abstractNote={Statistical information has been used extensively to define feasible solutions in set theoretic signal processing. However, the reliability of the set theoretic estimates resulting from the combination of multiple statistical constraints has thus far not been investigated. We address this question in order to provide a basis for a better use of statistical information in set theoretic estimation problems.}, number={3}, journal={IEEE Signal Processing Letters}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L. and Chaussalet, T.J.}, year={1996}, month={Mar}, pages={61–62} }
 @inproceedings{puh_combettes_1996, title={Set theoretic vector quantization}, booktitle={Proceedings of the Ninth IEEE Workshop on Image and Multidimensional Signal Processing}, author={Puh, H. and Combettes, P.L.}, year={1996}, pages={48–49} }
 @inbook{combettes_1996, title={The Convex Feasibility Problem in Image Recovery}, ISBN={9780120147373}, ISSN={1076-5670}, url={http://dx.doi.org/10.1016/s1076-5670(08)70157-5}, DOI={10.1016/s1076-5670(08)70157-5}, abstractNote={Image recovery is a broad discipline that encompasses the large body of inverse problems, in which an image h is to be inferred from the observation of data x consisting of signals physically or mathematically related to it. Image restoration and image reconstruction are the two main sub-branches of image recovery. The term “image restoration” usually applies to the problem of estimating the original form h of a degraded image x. The following four basic elements are required to solve an image recovery problem: (1) a data formation model, (2) a priori information, (3) a recovery criterion, and (4) a solution method. The recovery criterion defines the class of images that are acceptable as solutions to the problem. It is chosen by the user on grounds that may include experience, compatibility with the available a priori knowledge, personal convictions on the best way to solve the problem, and ease of implementation. The traditional approach has been to use a criterion of optimality, which usually leads to a single best solution. An alternative approach is to use a criterion of feasibility, in which consistency with all prior information and the data defines a set of equally acceptable solutions. The solution method is a numerical algorithm that will produce a solution to the recovery problem—that is, an image that satisfies the recovery criterion. Modification can be made in two directions: in the conventional image recovery framework, one seeks to preserve the notion of an optimal solution, whereas in the set theoretic framework the emphasis is placed on feasibility.}, booktitle={Advances in Imaging and Electron Physics}, publisher={Elsevier}, author={Combettes, P.L.}, year={1996}, pages={155–270} }
 @article{pesquet_combettes_1996, title={Wavelet synthesis by alternating projections}, volume={44}, ISSN={1053-587X}, url={http://dx.doi.org/10.1109/78.489050}, DOI={10.1109/78.489050}, abstractNote={The coefficients of the quadrature mirror filters involved in orthonormal wavelet or wavelet packets signal decompositions are often chosen in an ad hoc manner. In order to adapt such a decomposition to the signal being analyzed, it may be pertinent to maximize an energy concentration criterion, which leads to a constrained optimization problem. A simple geometric interpretation of this problem is given, and an alternating projection method is proposed to solve it.}, number={3}, journal={IEEE Transactions on Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Pesquet, J.-C. and Combettes, P.L.}, year={1996}, month={Mar}, pages={728–732} }
 @inproceedings{combettes_bondon_1995, place={Hoboken, New Jersey}, title={Adaptive linear filtering with convex constraints}, volume={2}, ISBN={0780324315}, url={http://dx.doi.org/10.1109/icassp.1995.480496}, DOI={10.1109/icassp.1995.480496}, abstractNote={We address the problem of linear mean-square estimation with arbitrary convex constraints for dependent processes. Two algorithms are proposed and their convergence is established. The first algorithm, which is deterministic, covers the case of known correlation structures; the second, which is stochastic and adaptive, covers the case of unknown correlation structures. Since existing algorithms can handle at most one simple constraint this contribution is relevant to signal processing problems in which arbitrary convex inequality constraints are present.}, booktitle={1995 International Conference on Acoustics, Speech, and Signal Processing}, publisher={IEEE}, author={Combettes, P.L. and Bondon, P.}, year={1995}, month={May}, pages={1372–1375} }
 @inproceedings{combettes_1995, place={Hoboken, New Jersey}, title={Constrained image recovery in a product space}, volume={2}, ISBN={0780331222}, url={http://dx.doi.org/10.1109/icip.1995.537406}, DOI={10.1109/icip.1995.537406}, abstractNote={In image recovery a priori knowledge and the observed data give rise to constraints on the solutions. In general, the recovery problem can be posed as that of minimizing a pertinent cost function over the resulting feasibility set. In this paper we present a product space framework for solving such problems, which leads to simplified formulations and to efficient parallel algorithms. Feasibility problems, quadratic minimization problems, and convex minimization problems are discussed.}, booktitle={Proceedings., International Conference on Image Processing}, publisher={IEEE Comput. Soc. Press}, author={Combettes, P.L.}, year={1995}, month={Nov}, pages={25–28} }
 @article{combettes_1995, title={Construction d'un point fixe commun à une famille de contractions fermes}, volume={320}, number={11}, journal={Comptes Rendus de l'Académie des Sciences de Paris, Série I (Mathématique)}, author={Combettes, P.L.}, year={1995}, month={Jun}, pages={1385–1390} }
 @article{combettes_trussell_1995, title={Deconvolution with bounded uncertainty}, volume={9}, ISSN={0890-6327 1099-1115}, url={http://dx.doi.org/10.1002/acs.4480090103}, DOI={10.1002/acs.4480090103}, abstractNote={Abstract}, number={1}, journal={International Journal of Adaptive Control and Signal Processing}, publisher={Wiley}, author={Combettes, Patrick L. and Trussell, H. Joel}, year={1995}, month={Jan}, pages={3–17} }
 @inbook{combettes_1995, title={Restauration ensembliste d’images par itérations parallèles extrapolées de sous-gradients}, url={http://hdl.handle.net/2042/12207}, booktitle={Actes du Quinzième Colloque GRETSI}, author={Combettes, P.L.}, year={1995}, month={Sep}, pages={447–450} }
 @article{bondon_combettes_picinbono_1995, title={Volterra filtering and higher order whiteness}, volume={43}, ISSN={1053-587X}, url={http://dx.doi.org/10.1109/78.414788}, DOI={10.1109/78.414788}, abstractNote={Some properties of Volterra filtering are established. Finite-order, finite-horizon Volterra filtering is investigated as well as its asymptotic properties. Next, the concepts of Volterra unpredictability and uninterpolability lead to generalizations of the notion of white noise to higher orders. These generalizations are introduced and relations are established between them. >}, number={9}, journal={IEEE Transactions on Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Bondon, P. and Combettes, P.L. and Picinbono, B.}, year={1995}, pages={2209–2212} }
 @inproceedings{combettes_puh_1994, place={Hoboken, New Jersey}, title={A fast parallel projection algorithm for set theoretic image recovery}, volume={5}, ISBN={0780317750}, url={http://dx.doi.org/10.1109/icassp.1994.389385}, DOI={10.1109/icassp.1994.389385}, abstractNote={A new projection algorithm for convex set theoretic image recovery [reconstruction and restoration] is presented. This algorithm comprises all serial and parallel projection methods as particular cases and is straightforwardly implementable on concurrent processors. It proceeds by taking convex combinations of selected projections at each iteration and allows extrapolated relaxations far beyond the range [0,2] used in conventional algorithms. These extrapolated, iteration-dependent relaxations result in very fast convergence. Numerical results are provided which show that the proposed algorithm outperforms existing ones, in particular the popular cyclic method of projections onto convex sets [POCS].<<ETX>>}, booktitle={Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing}, publisher={IEEE}, author={Combettes, P.L. and Puh, H.}, year={1994}, month={Apr}, pages={473–476} }
 @inproceedings{combettes_1994, place={Hoboken, New Jersey}, title={Convex set theoretic image recovery via chaotic iterations of approximate projections}, volume={3}, ISBN={0818669527}, url={http://dx.doi.org/10.1109/icip.1994.413863}, DOI={10.1109/icip.1994.413863}, abstractNote={Solving a convex set theoretic image recovery problem amounts to finding a point in the intersection of closed and convex sets in a Hilbert space. Methods employing projections onto the individual sets to build a sequence converging to a point in their intersection have proven most useful to obtain set theoretic solutions. They are nonetheless sometimes difficult to implement because of the theoretical or numerical tedium associated with the computation of projections at each iteration. We propose a general parallel iterative method which processes chaotically approximate projections instead of exact ones. Weak and strong convergence results are presented and subgradient projection methods are discussed as a particular case.<<ETX>>}, booktitle={Proceedings of 1st International Conference on Image Processing}, publisher={IEEE Comput. Soc. Press}, author={Combettes, P.L.}, year={1994}, month={Nov}, pages={182–186} }
 @article{combettes_1994, title={Inconsistent signal feasibility problems: least-squares solutions in a product space}, volume={42}, ISSN={1053-587X}, url={http://dx.doi.org/10.1109/78.330356}, DOI={10.1109/78.330356}, abstractNote={Presents parallel projection methods to find least-squares solutions to inconsistent convex set theoretic signal synthesis problems. The problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space, where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the Cartesian product of the original sets. A solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space. The convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated. >}, number={11}, journal={IEEE Transactions on Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L.}, year={1994}, pages={2955–2966} }
 @article{combettes_puh_1994, title={Iterations of parallel convex projections in hilbert spaces}, volume={15}, ISSN={0163-0563 1532-2467}, url={http://dx.doi.org/10.1080/01630569408816563}, DOI={10.1080/01630569408816563}, abstractNote={The problem of finding a common point of closed and convex sets in a Hilbert space is considered. A general iterative method of parallel projections is presented, in which the current iterate is projected simultaneously onto selected sets and the new iterate is a relaxed convex combination of the projections. Weak and strong convergence results are established and the influence of the relaxation coefficients is discussed. Convergence to a least-squares solution when the sets do not intersect is also proved.}, number={3-4}, journal={Numerical Functional Analysis and Optimization}, publisher={Informa UK Limited}, author={Combettes, Patrick L. and Puh, Hong}, year={1994}, month={Jan}, pages={225–243} }
 @inproceedings{combettes_chaussalet_1994, place={Hoboken, New Jersey}, title={Selecting Statistical Information In Set Theoretic Signal Processing}, url={http://dx.doi.org/10.1109/ssap.1994.572433}, DOI={10.1109/ssap.1994.572433}, booktitle={IEEE Seventh SP Workshop on Statistical Signal and Array Processing}, publisher={IEEE}, author={Combettes, P.L. and Chaussalet, T.J.}, year={1994}, month={Jun}, pages={55–58} }
 @inproceedings{combettes_1994, place={Hoboken, New Jersey}, title={Set Theoretic Signal Processing}, url={http://dx.doi.org/10.1109/ssap.1994.572418}, DOI={10.1109/ssap.1994.572418}, booktitle={IEEE Seventh SP Workshop on Statistical Signal and Array Processing}, publisher={IEEE}, author={Combettes, P.L.}, year={1994}, month={Jun}, pages={1–6} }
 @inproceedings{pesquet_combettes_1994, title={Synthèse ensembliste d’ondelettes}, booktitle={Actes de la Conférence Temps-Fréquence, Ondelettes et Multirésolution}, author={Pesquet, J.C. and Combettes, P. L.}, year={1994}, month={Mar}, pages={14.1–14.10} }
 @inproceedings{combettes_1993, title={A simultaneous projection method for inconsistent signal and image feasibility problems}, booktitle={Proceedings of the Eighth IEEE Workshop on Image and Multidimensional Signal Processing}, author={Combettes, P.L.}, year={1993}, pages={32–33} }
 @inbook{combettes_chaussalet_1993, title={Estimation en présence de modèles incertains: sélection de formulations ensemblistes}, url={http://hdl.handle.net/2042/12154}, booktitle={Actes du Quatorzième Colloque GRETSI}, author={Combettes, P.L. and Chaussalet, T.J.}, year={1993}, month={Sep}, pages={205–208} }
 @inproceedings{combettes_puh_1993, place={Hoboken, New Jersey}, title={Parallel projection methods for set theoretic signal reconstruction and restoration}, ISBN={0780309464}, url={http://dx.doi.org/10.1109/icassp.1993.319806}, DOI={10.1109/icassp.1993.319806}, abstractNote={An attempt is made to lay a theoretical and computational foundation for the use of parallel projection methods in set-theoretic signal restoration and reconstruction. A general method of parallel projections (MOPP) for solving feasibility problems in Hilbert spaces is proposed. In the proposed scheme, an elementary iteration consists in projecting the current estimate simultaneously onto selected sets and forming a relaxed convex combination of the projections. MOPP is seen to have substantial advantages over the method of successive projections, the widely used serial projection method which it generalizes: straightforward implementation on a parallel computing architecture, an explicit condition on the relaxation coefficients for faster convergence, and convergence to weighted least-squares solutions for inconsistent set-theoretic formulations. Convergence results for this algorithm in the convex and nonconvex cases are presented. Both consistent and inconsistent set-theoretic formulations are considered. Practical issues pertaining to the utilization of the method are discussed.<<ETX>>}, booktitle={IEEE International Conference on Acoustics Speech and Signal Processing}, publisher={IEEE}, author={Combettes, P.L. and Puh, H.}, year={1993} }
 @article{combettes_1993, title={Signal recovery by best feasible approximation}, volume={2}, ISSN={1057-7149}, url={http://dx.doi.org/10.1109/83.217232}, DOI={10.1109/83.217232}, abstractNote={The objective of set theoretical signal recovery is to find a feasible signal in the form of a point in the intersection of S of sets modeling the information available about the problem. For problems in which the true signal is known to lie near a reference signal r, the solution should not be any feasible point but one which best approximates r, i.e., a projection of r onto S. Such a solution cannot be obtained by the feasibility algorithms currently in use, e.g., the method of projections onto convex sets (POCS) and its offsprings. Methods for projecting a point onto the intersection of closed and convex sets in a Hilbert space are introduced and applied to signal recovery by best feasible approximation of a reference signal. These algorithms are closely related to the above projection methods, to which they add little computational complexity.}, number={2}, journal={IEEE Transactions on Image Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L.}, year={1993}, month={Apr}, pages={269–271} }
 @article{combettes_1993, title={The foundations of set theoretic estimation}, volume={81}, ISSN={0018-9219}, url={http://dx.doi.org/10.1109/5.214546}, DOI={10.1109/5.214546}, abstractNote={Explains set theoretic estimation, which is governed by the notion of feasibility and produces solutions whose sole property is to be consistent with all information arising from the observed data and a priori knowledge. Each piece of information is associated with a set in the solution space, and the intersection of these sets, the feasibility set, represents the acceptable solutions. The practical use of the set theoretic framework stems from the existence of efficient techniques for finding these solutions. Many scattered problems in systems science and signal processing have been approached in set theoretic terms over the past three decades. The author synthesizes a single, general framework from these various approaches, examines its fundamental philosophy, goals, and analytical techniques, and relates it to conventional methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>}, number={2}, journal={Proceedings of the IEEE}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L.}, year={1993}, month={Feb}, pages={182–208} }
 @inproceedings{bondon_combettes_picinbono_1993, place={Hoboken, New Jersey}, title={Volterra prediction models and higher order whiteness}, ISBN={0780309464}, url={http://dx.doi.org/10.1109/icassp.1993.319632}, DOI={10.1109/icassp.1993.319632}, abstractNote={For nonGaussian processes, a nonlinear predictor can achieve a smaller prediction error than a linear one. The authors study Volterra predictors and compare them with their linear counterparts. The concept of Volterra unpredictability leads to important generalizations of the notion of white noise to higher orders. These generalizations are introduced and relations are established between them. Examples are provided to show that the relations between these higher order whitenesses are not trivial.<<ETX>>}, booktitle={IEEE International Conference on Acoustics Speech and Signal Processing}, publisher={IEEE}, author={Bondon, P. and Combettes, P.L. and Picinbono, B.}, year={1993} }
 @article{zilovic_roytman_combettes_swamy_1992, title={A bound for the zeros of polynomials}, volume={39}, ISSN={1057-7122}, url={http://dx.doi.org/10.1109/81.153643}, DOI={10.1109/81.153643}, abstractNote={A new kind of circular bound on the zeros of polynomials is derived by determining Cauchy's bound on zeros of its transformed pair first. The transformation is based on a nonlinear transformation of the variable, which conceptually should give a better upper bound. The advantage of such a transformation is illustrated through several examples that show the improvement over the existing bounds. Convergence of the bound after iterative transformations of the original polynomial is also examined. >}, number={6}, journal={IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Zilovic, M.S. and Roytman, L.M. and Combettes, P.L. and Swamy, M.N.S.}, year={1992}, month={Jun}, pages={476–478} }
 @inproceedings{combettes_benidir_picinbono_1992, place={Hoboken, New Jersey}, title={A general framework for the incorporation of uncertainty in set theoretic estimation}, ISBN={0780305329}, url={http://dx.doi.org/10.1109/icassp.1992.226229}, DOI={10.1109/icassp.1992.226229}, abstractNote={In digital signal processing, the two main sources of uncertainty encountered in estimation problems are model uncertainty and noise. In many instances, probabilistic information is available to partially describe these sources of uncertainty. It is shown how such information can be exploited in a broad class of set theoretic estimation problems relevant to digital signal processing. A general framework is developed to construct sets in the solution space by constraining the estimation residual based on the known component of the model to be consistent with those known properties of a so-called uncertainty process consisting of the contribution of the unknown component of the model and the noise. Specific digital signal processing applications are discussed.<<ETX>>}, booktitle={ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing}, publisher={IEEE}, author={Combettes, P.L. and Benidir, M. and Picinbono, B.}, year={1992} }
 @article{combettes_trussell_1992, title={Best stable and invertible approximations for ARMA systems}, volume={40}, ISSN={1053-587X}, url={http://dx.doi.org/10.1109/78.175751}, DOI={10.1109/78.175751}, abstractNote={A method is proposed for finding the best stable and invertible approximations for an autoregressive moving average (ARMA) system, relative to a general quadratic metric in the coefficient space. Mathematically, the problem is equivalent to projecting the regression and moving average vectors of the system onto the set S of coefficients of monic Schur polynomials. The geometry of S is too complex to allow the problem to be approached directly in the ARMA coefficient space. A solution is obtained by constrained steepest descent in the hypercube of reflection coefficients, which is homomorphic to S. >}, number={12}, journal={IEEE Transactions on Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L. and Trussell, H.J.}, year={1992}, pages={3066–3069} }
 @article{combettes_1992, title={Convex set theoretic image recovery: History, current status, and new directions}, volume={3}, ISSN={1047-3203}, url={http://dx.doi.org/10.1016/1047-3203(92)90034-q}, DOI={10.1016/1047-3203(92)90034-q}, abstractNote={Since its introduction a decade ago, convex set theoretic image recovery has been applied to a steadily increasing number of problems and has established itself as a powerful, flexible, and reliable estimation tool. The purpose of this paper is to give a historical overview of the field, survey its most significant developments, analyze its current limitations, and propose new directions for theoretical and applied research.}, number={4}, journal={Journal of Visual Communication and Image Representation}, publisher={Elsevier BV}, author={Combettes, Patrick L.}, year={1992}, month={Dec}, pages={307–315} }
 @inproceedings{silverstein_combettes_1992, place={Hoboken, New Jersey}, title={Large dimensional random matrix theory for signal detection and estimation in array processing}, ISBN={0780305086}, url={http://dx.doi.org/10.1109/ssap.1992.246796}, DOI={10.1109/ssap.1992.246796}, abstractNote={This paper brings into play elements of the spectral theory of such matrices and demonstrates their relevance to source detection and bearing estimation in problems with sizable arrays. These results are applied to the sample spatial covariance matrix, R, of the sensed data. It is seen that detection can be achieved with a sample size considerably less than that required by conventional approaches. It is argued that more accurate estimates of direction of arrival can be obtained by constraining R to be consistent with various a priori constraints including those arising from large dimensional random matrix theory. A set theoretic formalism is used for this feasibility problem. Unsolved issues are discussed.<<ETX>>}, booktitle={IEEE Sixth SP Workshop on Statistical Signal and Array Processing}, publisher={IEEE}, author={Silverstein, J.W. and Combettes, P.L.}, year={1992}, month={Oct}, pages={276–279} }
 @article{silverstein_combettes_1992, title={Signal detection via spectral theory of large dimensional random matrices}, volume={40}, ISSN={1053-587X}, url={http://dx.doi.org/10.1109/78.149981}, DOI={10.1109/78.149981}, abstractNote={Results on the spectral behavior of random matrices as the dimension increases are applied to the problem of detecting the number of sources impinging on an array of sensors. A common strategy to solve this problem is to estimate the multiplicity of the smallest eigenvalue of the spatial covariance matrix R of the sensed data. Existing approaches rely on the closeness of the noise eigenvalues of sample covariance matrix to each other and, therefore, the sample size has to be quite large when the number of sources is large in order to obtain a good estimate. The theoretical analysis presented focuses on the splitting of the spectrum of sample covariance matrix into noise and signal eigenvalues. It is shown that when the number of sensors is large the number of signals can be estimated with a sample size considerably less than that required by previous approaches. >}, number={8}, journal={IEEE Transactions on Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Silverstein, J.W. and Combettes, P.L.}, year={1992}, pages={2100–2105} }
 @inproceedings{combettes_edmonson_1992, title={What is a good estimate?}, booktitle={Proceedings of the European Signal Processing Conference}, author={Combettes, P.L. and Edmonson, W.W.}, year={1992}, pages={713–716} }
 @inproceedings{combettes_chaussalet_1991, title={Critères de qualité en estimation ensembliste}, booktitle={Actes du Treizième Colloque GRETSI}, publisher={Juan-les-Pins}, author={Combettes, P.L. and Chaussalet, T.J.}, year={1991}, month={Sep}, pages={249–252} }
 @article{combettes_trussell_1991, title={Set theoretic estimation by random search}, volume={39}, ISSN={1053-587X}, url={http://dx.doi.org/10.1109/78.134403}, DOI={10.1109/78.134403}, abstractNote={An adapted random search algorithm is shown to be a feasible method for the synthesis of set theoretic estimates. It circumvents the theoretical and computational shortcomings of existing deterministic methods and does not place any geometrical restrictions on the sets. The proposed method can handle arbitrarily complex property sets in applications for which the number of unknown parameters is typically low (e.g. parametric multidimensional spectral estimation, system identification, blur identification, and filter design). >}, number={7}, journal={IEEE Transactions on Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L. and Trussell, H.J.}, year={1991}, month={Jul}, pages={1669–1671} }
 @inproceedings{combettes_civanlar_1991, title={The foundations of set theoretic estimation}, ISBN={0780300033}, url={http://dx.doi.org/10.1109/icassp.1991.151014}, DOI={10.1109/icassp.1991.151014}, abstractNote={Many scattered estimation problems in systems science and signal processing have been approached in set theoretic terms over the past three decades. A single formal framework is presented to synthesize these various approaches, and the fundamental philosophy, goals, and analytical techniques of set theoretic estimation are discussed.<<ETX>>}, booktitle={[Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing}, publisher={IEEE}, author={Combettes, P.L. and Civanlar, M.R.}, year={1991} }
 @article{combettes_trussell_1991, title={The use of noise properties in set theoretic estimation}, volume={39}, ISSN={1053-587X}, url={http://dx.doi.org/10.1109/78.134400}, DOI={10.1109/78.134400}, abstractNote={In most digital signal processing problems, the goal is to estimate an object from noise corrupted observations of a physical system. The authors describe how a wide range of probabilistic information pertaining to the noise process can be used in a general set theoretic estimation framework. The basic principle is to constrain the sample statistics of the estimation residual to be consistent with those probabilistic properties of the noise which are available and to construct sets accordingly in the solution space. Adding these sets to the collection of sets describing the solution will yield a smaller feasibility set and, hence, more reliable estimates. Pieces of information relative to quantities such as range, moments, absolute moments, and second and higher order probabilistic attributes are considered, and properties of the corresponding sets are established. Simulations are provided to illustrate the theoretical developments. >}, number={7}, journal={IEEE Transactions on Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L. and Trussell, H.J.}, year={1991}, month={Jul}, pages={1630–1641} }
 @article{combettes_trussell_1990, title={Method of successive projections for finding a common point of sets in metric spaces}, volume={67}, ISSN={0022-3239 1573-2878}, url={http://dx.doi.org/10.1007/bf00939646}, DOI={10.1007/bf00939646}, number={3}, journal={Journal of Optimization Theory and Applications}, publisher={Springer Science and Business Media LLC}, author={Combettes, P. L. and Trussell, H. J.}, year={1990}, month={Dec}, pages={487–507} }
 @inproceedings{combettes_trussell_1990, place={Hoboken, New Jersey}, title={New methods for the synthesis of set theoretic estimates (digital signal processing)}, url={http://dx.doi.org/10.1109/icassp.1990.116116}, DOI={10.1109/icassp.1990.116116}, abstractNote={Two methods for the synthesis of set theoretic estimates are presented. The first is a generalization of the method of successive projections onto closed and convex subsets of Hilbert spaces to approximately compact subsets of metric spaces. The second is based on a stochastic search in the solution space. These methods allow greater flexibility with regard to the incorporation of prior knowledge, thereby extending the scope of set theoretic estimation. Applications to digital signal processing problems are discussed.<<ETX>>}, booktitle={International Conference on Acoustics, Speech, and Signal Processing}, publisher={IEEE}, author={Combettes, P.L. and Trussell, H.J.}, year={1990}, month={Apr}, pages={2531–2534} }
 @inproceedings{combettes_trussell_1990, place={Hoboken, New Jersey}, title={Set theoretic autoregressive spectral estimation}, url={http://dx.doi.org/10.1109/spect.1990.205587}, DOI={10.1109/spect.1990.205587}, abstractNote={Presents the set theoretic approach in autoregressive (AR) spectral estimation. Conventional AR estimates, which are based on some criterion of optimality, may violate a priori constraints on the problem. In the framework of set theoretic estimation, one produces an estimate of the regression vector which has the property of being consistent with all the available a priori knowledge. Each known property being associated with a set in the regression space, the problem is then to find a common point of these sets.<<ETX>>}, booktitle={Fifth ASSP Workshop on Spectrum Estimation and Modeling}, publisher={IEEE}, author={Combettes, P.L. and Trussell, H.J.}, year={1990} }
 @inproceedings{combettes_trussell_1989, place={Hoboken, New Jersey}, title={General order moments in set theoretic estimation}, url={http://dx.doi.org/10.1109/icassp.1989.266876}, DOI={10.1109/icassp.1989.266876}, abstractNote={A description is given of how information pertaining to an arbitrary absolute moment of the noise process can be used in a general set-theoretic estimation framework. It is shown that for each such piece of a priori information, a set can be constructed in the solution space by constraining the corresponding sample statistics of the estimation residual to lie within some acceptable distance from the expected value. The geometrical and topological properties of these sets are investigated. The benefits of the simultaneous use of several such sets with regard to the synthesis of better set-theoretic estimates are discussed. Applications to signal restoration/reconstruction, speech prediction, spectral estimation, and radar signal processing are discussed.<<ETX>>}, booktitle={International Conference on Acoustics, Speech, and Signal Processing}, publisher={IEEE}, author={Combettes, P.L. and Trussell, H.J.}, year={1989}, month={May}, pages={2531–2534} }
 @article{combettes_trussell_1989, title={Methods for digital restoration of signals degraded by a stochastic impulse response}, volume={37}, ISSN={0096-3518}, url={http://dx.doi.org/10.1109/29.21706}, DOI={10.1109/29.21706}, abstractNote={Several digital restoration methods are extended to the cases in which signals have suffered stochastic blurring. Rather than being regarded in terms of averages, these degradations are treated dynamically in the derivation of restoration algorithms based on various mathematical criteria. It is shown that the integration of the additional uncertainties caused by the variations of the impulse response of the stochastic degradation can be used to obtain better estimates. These uncertainties can be included in the restoration scheme in a very flexible manner through the use of the projection-onto-convex-sets (POCS) method. Conventional constrained methods are also addressed. The benefits of the method are illustrated through one- and two-dimensional simulations. >}, number={3}, journal={IEEE Transactions on Acoustics, Speech, and Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Combettes, P.L. and Trussell, H.J.}, year={1989}, month={Mar}, pages={393–401} }
 @inproceedings{combettes_trussell_1988, place={Hoboken, New Jersey}, title={Stability of the linear prediction filter: a set theoretic approach}, url={http://dx.doi.org/10.1109/icassp.1988.197094}, DOI={10.1109/icassp.1988.197094}, abstractNote={The authors investigate the application of set-theoretic estimation methods to the design of linear-predictive-coding (LPC) digital filters. The constraints involved in the design are represented by sets in an abstract space. Any member of their intersection satisfies all the design specifications and is called feasible. Two sets are considered: the set of stable filters and the set of filters which minimize the mean-square prediction error in an environment where the statistics of the observed data are unknown. These two sets are described analytically and the problem of obtaining a feasible LPC filter is addressed. Experimental results are also presented.<<ETX>>}, booktitle={ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing}, publisher={IEEE}, author={Combettes, P.L. and Trussell, H.J.}, year={1988}, month={Apr}, pages={2288–2291} }
 @inproceedings{trussell_combettes_1987, place={Hoboken, New Jersey}, title={Considerations for the restoration of stochastic degradations}, url={http://dx.doi.org/10.1109/icassp.1987.1169769}, DOI={10.1109/icassp.1987.1169769}, abstractNote={This paper presents a technique for restoring images which have been degraded by a stochastic point spread function (psf). In the past, the restoration of such images has been approached in terms of averages without considering the variations of the psf. It will be shown that the integration of the additional uncertainties caused by the stochastic psf can be used to obtain better estimates. These uncertainties can be included to the restoration scheme in a very flexible manner through the use of the Projection Onto Convex Sets (POCS) method. Finally, the benefits of the method will be illustrated through simulations.}, booktitle={ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing}, publisher={Institute of Electrical and Electronics Engineers}, author={Trussell, H. and Combettes, P.}, year={1987}, month={Apr}, pages={1209–1212} }
 @inproceedings{combettes_trussell_1987, title={Modèles et algorithmes en vue de la restauration numérique d’images rayons-X}, booktitle={Actes du Colloque MARI-Cognitiva Electronic Image}, author={Combettes, P.L. and Trussell, H.J.}, year={1987}, month={May}, pages={146–151} }