@article{murray_wilcox_2023, title={NON-LINEAR SINGULARITY FORMATION FOR CIRCULAR VORTEX SHEETS}, volume={5}, ISSN={["1552-4485"]}, DOI={10.1090/qam/1659}, abstractNote={We study the evolution of vortex sheets according to the Birkhoff-Rott equation, which describe the motion of sharp shear interfaces governed by the incompressible Euler equation in two dimensions. In a recent work, the authors demonstrated within this context a marginal linear stability of circular vortex sheets, standing in sharp contrast with classical instability of the flat vortex sheet, which is known as the Kelvin-Helmholtz instability. This article continues that analysis by investigating how non-linear effects induce singularity formation near the circular vortex sheet. In high-frequency regimes, the singularity formation is primarily driven by a complex-valued, conjugated Burgers equation, which we study by modifying a classical argument from hyperbolic conservation laws. This provides a deeper understanding of the mechanisms driving the breakdown of circular vortex sheets, which are observed both numerically and experimentally.}, journal={QUARTERLY OF APPLIED MATHEMATICS}, author={Murray, Ryan and Wilcox, Galen}, year={2023}, month={May} }
@article{bungert_trillos_murray_2023, title={The geometry of adversarial training in binary classification}, volume={1}, ISSN={["2049-8772"]}, DOI={10.1093/imaiai/iaac029}, abstractNote={Abstract We establish an equivalence between a family of adversarial training problems for non-parametric binary classification and a family of regularized risk minimization problems where the regularizer is a nonlocal perimeter functional. The resulting regularized risk minimization problems admit exact convex relaxations of the type $L^1+\text{(nonlocal)}\operatorname{TV}$, a form frequently studied in image analysis and graph-based learning. A rich geometric structure is revealed by this reformulation which in turn allows us to establish a series of properties of optimal solutions of the original problem, including the existence of minimal and maximal solutions (interpreted in a suitable sense) and the existence of regular solutions (also interpreted in a suitable sense). In addition, we highlight how the connection between adversarial training and perimeter minimization problems provides a novel, directly interpretable, statistical motivation for a family of regularized risk minimization problems involving perimeter/total variation. The majority of our theoretical results are independent of the distance used to define adversarial attacks.}, journal={INFORMATION AND INFERENCE-A JOURNAL OF THE IMA}, author={Bungert, Leon and Trillos, Nicolas Garcia and Murray, Ryan}, year={2023}, month={Jan} }
@article{adversarial classification: necessary conditions and geometric flows_2022, volume={23}, number={187}, journal={Journal of Machine Learning Research}, year={2022}, pages={1–38} }
@article{swenson_murray_poor_kar_2022, title={Distributed Gradient Flow: Nonsmoothness, Nonconvexity, and Saddle Point Evasion}, volume={67}, ISSN={["1558-2523"]}, url={https://doi.org/10.1109/TAC.2021.3111853}, DOI={10.1109/TAC.2021.3111853}, abstractNote={The article considers distributed gradient flow (DGF) for multiagent nonconvex optimization. DGF is a continuous-time approximation of distributed gradient descent that is often easier to study than its discrete-time counterpart. The article has two main contributions. First, the article considers optimization of nonsmooth, nonconvex objective functions. It is shown that DGF converges to critical points in this setting. The article then considers the problem of avoiding saddle points. It is shown that if agents’ objective functions are assumed to be smooth and nonconvex, then DGF can only converge to a saddle point from a zero-measure set of initial conditions. To establish this result, the article proves a stable manifold theorem for DGF, which is a fundamental contribution of independent interest. In a companion article, analogous results are derived for discrete-time algorithms.}, number={8}, journal={IEEE TRANSACTIONS ON AUTOMATIC CONTROL}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Swenson, Brian and Murray, Ryan and Poor, H. Vincent and Kar, Soummya}, year={2022}, month={Aug}, pages={3949–3964} }
@article{trillos_murray_thorpe_2022, title={From Graph Cuts to Isoperimetric Inequalities: Convergence Rates of Cheeger Cuts on Data Clouds}, volume={4}, ISSN={["1432-0673"]}, DOI={10.1007/s00205-022-01770-8}, abstractNote={Abstract In this work we study statistical properties of graph-based clustering algorithms that rely on the optimization of balanced graph cuts, the main example being the optimization of Cheeger cuts. We consider proximity graphs built from data sampled from an underlying distribution supported on a generic smooth compact manifold $${\mathcal {M}}$$ M . In this setting, we obtain high probability convergence rates for both the Cheeger constant and the associated Cheeger cuts towards their continuum counterparts. The key technical tools are careful estimates of interpolation operators which lift empirical Cheeger cuts to the continuum, as well as continuum stability estimates for isoperimetric problems. To the best of our knowledge the quantitative estimates obtained here are the first of their kind.}, journal={ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS}, author={Trillos, Nicolas Garcia and Murray, Ryan and Thorpe, Matthew}, year={2022}, month={Apr} }
@article{molina-fructuoso_murray_2022, title={Tukey Depths and Hamilton-Jacobi Differential Equations}, volume={4}, ISSN={["2577-0187"]}, url={https://doi.org/10.1137/21M1411998}, DOI={10.1137/21M1411998}, abstractNote={Widespread application of modern machine learning has increased the need for robust statistical algorithms. This work studies one such fundamental statistical concept known as the Tukey depth. We study the problem in the continuum (population) limit. In particular, we formally derive the associated necessary conditions, which take the form of a first-order partial differential equation which is necessarily satisfied at points where the Tukey depth is smooth. We discuss the interpretation of this formal necessary condition in terms of the viscosity solution of a Hamilton--Jacobi equation, but with a nonclassical Hamiltonian with discontinuous dependence on the gradient at zero. We prove that this equation possesses a unique viscosity solution and that this solution always bounds the Tukey depth from below. In certain cases we prove that the Tukey depth is equal to the viscosity solution, and we give some illustrations of standard numerical methods from the optimal control community which deal directly with the partial differential equation. We conclude by outlining several promising research directions both in terms of new numerical algorithms and theoretical challenges.}, number={2}, journal={SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE}, author={Molina-Fructuoso, Martin and Murray, Ryan}, year={2022}, pages={604–633} }
@article{murray_fokoue_2021, title={Dropout Fails to Regularize Nonparametric Learners}, volume={15}, ISSN={["1559-8616"]}, url={https://doi.org/10.1007/s42519-020-00158-9}, DOI={10.1007/s42519-020-00158-9}, number={2}, journal={JOURNAL OF STATISTICAL THEORY AND PRACTICE}, publisher={Springer Science and Business Media LLC}, author={Murray, Ryan W. and Fokoue, Ernest}, year={2021}, month={Jan} }
@article{murray_young_2020, title={Neutral competition in a deterministically changing environment: Revisiting continuum approaches}, volume={486}, ISSN={["1095-8541"]}, DOI={10.1016/j.jtbi.2019.110104}, abstractNote={Environmental variation can play an important role in ecological competition by influencing the relative advantage between competing species. Here, we consider such effects by extending a classical, competitive Moran model to incorporate an environment that fluctuates periodically in time. We adapt methods from work on these classical models to investigate the effects of the magnitude and frequency of environmental fluctuations on two important population statistics: the probability of fixation and the mean time to fixation. In particular, we find that for small frequencies, the system behaves similar to a system with a constant fitness difference between the two species, and for large frequencies, the system behaves similar to a neutrally competitive model. Most interestingly, the system exhibits nontrivial behavior for intermediate frequencies. We conclude by showing that our results agree quite well with recent theoretical work on competitive models with a stochastically changing environment, and discuss how the methods we develop ease the mathematical analysis required to study such models.}, journal={JOURNAL OF THEORETICAL BIOLOGY}, author={Murray, Ryan and Young, Glenn}, year={2020}, month={Feb} }
@article{bressan_murray_2020, title={On self-similar solutions to the incompressible Euler equations}, volume={269}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2020.04.005}, abstractNote={Recent numerical simulations have shown the existence of multiple self-similar solutions to the Cauchy problem for the 2-dimensional incompressible Euler equation, with initial vorticity in Llocp(R2), 1≤p<+∞. Toward a rigorous validation of these computations, in this paper we analytically construct self-similar solutions (i) on an outer domain of the form {|x|>R}, and (ii) in a neighborhood of the points where the solution exhibits a spiraling vortex singularity. The outer solution is obtained as the fixed point of a contractive transformation, based on the Biot-Savart formula and integration along characteristics. The inner solution is constructed using a system of adapted coordinates, following the approach of V. Elling (2016) [17].}, number={6}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Bressan, Alberto and Murray, Ryan}, year={2020}, month={Sep}, pages={5142–5203} }
@article{swenson_murray_kar_2020, title={Regular potential games}, volume={124}, url={https://doi.org/10.1016/j.geb.2020.09.005}, DOI={10.1016/j.geb.2020.09.005}, abstractNote={Abstract A fundamental problem with the Nash equilibrium concept is the existence of certain “structurally deficient” equilibria that (i) lack fundamental robustness properties, and (ii) are difficult to analyze. The notion of a “regular” Nash equilibrium was introduced by Harsanyi. Such equilibria are isolated, highly robust, and relatively simple to analyze. A game is said to be regular if all equilibria in the game are regular. In this paper it is shown that almost all potential games are regular. That is, except for a closed subset with Lebesgue measure zero, all potential games are regular. As an immediate consequence of this, the paper also proves an oddness result for potential games: In almost all potential games, the number of Nash equilibrium strategies is finite and odd. Specialized results are given for weighted potential games, exact potential games, and games with identical payoffs. Applications of the results to game-theoretic learning are discussed.}, journal={Games and Economic Behavior}, publisher={Elsevier BV}, author={Swenson, Brian and Murray, Ryan and Kar, Soummya}, year={2020}, month={Nov}, pages={432–453} }
@article{trillos_murray_2020, title={y A Maximum Principle Argument for the Uniform Convergence of Graph Laplacian Regressors}, volume={2}, ISSN={["2577-0187"]}, url={https://doi.org/10.1137/19M1245372}, DOI={10.1137/19M1245372}, abstractNote={This paper investigates the use of methods from partial differential equations and the calculus of variations to study learning problems that are regularized using graph Laplacians. Graph Laplacians are a powerful, flexible method for capturing local and global geometry in many classes of learning problems, and the techniques developed in this paper help to broaden the methodology of studying such problems. In particular, we develop the use of maximum principle arguments to establish asymptotic consistency guarantees within the context of noise corrupted, nonparametric regression with samples living on an unknown manifold embedded in $\mathbb{R}^d$. The maximum principle arguments provide a new technical tool which informs parameter selection by giving concrete error estimates in terms of various regularization parameters. A review of learning algorithms which utilize graph Laplacians, as well as previous developments in the use of differential equation and variational techniques to study those algorithms, is given. In addition, new connections are drawn between Laplacian methods and other machine learning techniques, such as kernel regression and $k$-nearest neighbor methods.}, number={3}, journal={SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Trillos, Nicolas Garcia and Murray, Ryan W.}, year={2020}, pages={705–739} }
@article{leoni_murray_2019, title={Local minimizers and slow motion for the mass preserving Allen–Cahn equation in higher dimensions}, volume={147}, DOI={10.1090/proc/13988}, abstractNote={This paper completely resolves the asymptotic development of order 2 2 by Γ \Gamma -convergence of the mass-constrained Cahn–Hilliard functional. Important new results on the slow motion of interfaces for the mass preserving Allen–Cahn equation and the Cahn–Hilliard equations in higher dimension are obtained as an application.}, number={12}, journal={Proceedings of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Leoni, Giovanni and Murray, Ryan}, year={2019}, month={Sep}, pages={5167–5182} }
@article{murray_swenson_kar_2019, title={Revisiting Normalized Gradient Descent: Fast Evasion of Saddle Points}, volume={64}, url={https://doi.org/10.1109/TAC.2019.2914998}, DOI={10.1109/TAC.2019.2914998}, abstractNote={The paper considers normalized gradient descent (NGD), a natural modification of classical gradient descent (GD) in optimization problems. It is shown that, contrary to GD, NGD escapes saddle points “quickly.” A serious shortcoming of GD in nonconvex problems is that it can take arbitrarily long to escape from the neighborhood of a saddle point. In practice, this issue can significantly slow the convergence of GD, particularly in high-dimensional nonconvex problems. The paper focuses on continuous-time dynamics. It is shown that 1) NGD “almost never” converges to saddle points and 2) the time required for NGD to escape from a ball of radius r about a saddle point x* is at most 5√(κr), where κ is the condition number of the Hessian of f at x*. As a simple application of these results, a global convergence-time bound is established for NGD under mild assumptions.}, number={11}, journal={IEEE Transactions on Automatic Control}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Murray, Ryan and Swenson, Brian and Kar, Soummya}, year={2019}, month={Nov}, pages={4818–4824} }
@article{murray_palladino_2018, title={A model for system uncertainty in reinforcement learning}, volume={122}, url={https://doi.org/10.1016/j.sysconle.2018.09.011}, DOI={10.1016/j.sysconle.2018.09.011}, abstractNote={Abstract This work provides a rigorous framework for studying continuous-time control problems in uncertain environments. The framework models uncertainty in state dynamics as a probability measure on the space of functions. Such a probability measure is permitted to change over time as agents learn about their environment. This model can be seen as a variant of either Bayesian reinforcement learning (RL) or adaptive optimal control. We study conditions for locally optimal trajectories within this model, in particular deriving an appropriate dynamic programming principle and Hamilton–Jacobi equations. Some discussion of variants of the model are also provided, including one potential framework for studying the tradeoff between exploration and exploitation in RL.}, journal={Systems & Control Letters}, publisher={Elsevier BV}, author={Murray, Ryan and Palladino, Michele}, year={2018}, month={Dec}, pages={24–31} }
@article{swenson_murray_kar_poor_2018, title={Best-Response Dynamics in Continuous Potential Games: Non-Convergence to Saddle Points}, DOI={10.1109/acssc.2018.8645541}, abstractNote={The paper studies properties of best-response (BR) dynamics in potential games with continuous action sets. It is known that BR dynamics converge to the set of Nash equilibria (NE) in potential games. The set of NE in potential games is composed of local maximizers and saddle points of the potential function. The paper studies non-convergence of BR dynamics to saddle points of the potential function. Under relatively mild assumptions it is shown that BR dynamics may only converge to an interior saddle-point from a measure-zero set of initial conditions. This provides a weak stable manifold theorem in this context.}, journal={2018 52nd Asilomar Conference on Signals, Systems, and Computers}, publisher={IEEE}, author={Swenson, Brian and Murray, Ryan and Kar, Soummya and Poor, H. Vincent}, year={2018}, month={Oct} }
@article{swenson_murray_kar_2018, title={On Best-Response Dynamics in Potential Games}, volume={56}, DOI={10.1137/17m1139461}, abstractNote={The paper studies the convergence properties of (continuous-time) best-response dynamics from game theory. Despite their fundamental role in game theory, best-response dynamics are poorly understoo...}, number={4}, journal={SIAM Journal on Control and Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Swenson, Brian and Murray, Ryan and Kar, Soummya}, year={2018}, month={Jan}, pages={2734–2767} }
@article{trillos_murray_2017, title={A new analytical approach to consistency and overfitting in regularized empirical risk minimization}, volume={28}, DOI={10.1017/s0956792517000201}, abstractNote={This work considers the problem of binary classification: given training data x 1 , . . ., x n from a certain population, together with associated labels y 1 ,. . ., y n ∈ {0,1}, determine the best label for an element x not among the training data. More specifically, this work considers a variant of the regularized empirical risk functional which is defined intrinsically to the observed data and does not depend on the underlying population. Tools from modern analysis are used to obtain a concise proof of asymptotic consistency as regularization parameters are taken to zero at rates related to the size of the sample. These analytical tools give a new framework for understanding overfitting and underfitting, and rigorously connect the notion of overfitting with a loss of compactness.}, number={6}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={TRILLOS, NICOLÁS GARCÍA and MURRAY, RYAN}, year={2017}, month={Jul}, pages={886–921} }
@article{murray_pego_2017, title={Cutoff estimates for the linearized Becker–Döring equations}, volume={15}, DOI={10.4310/cms.2017.v15.n6.a10}, number={6}, journal={Communications in Mathematical Sciences}, publisher={International Press of Boston}, author={Murray, Ryan W. and Pego, Robert L.}, year={2017}, pages={1685–1702} }
@article{murray_pego_2016, title={Algebraic Decay to Equilibrium for the Becker--Döring Equations}, volume={48}, DOI={10.1137/15m1038578}, abstractNote={This paper studies rates of decay to equilibrium for the Becker--Döring equations with subcritical initial data. In particular, algebraic rates of decay are established when initial perturbations of equilibrium have polynomial moments. This is proved by using new dissipation estimates in polynomially weighted $\ell^1$ spaces, operator decomposition techniques from kinetic theory, and interpolation estimates from the study of traveling waves.}, number={4}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Murray, Ryan W. and Pego, Robert L.}, year={2016}, month={Jan}, pages={2819–2842} }
@article{leoni_murray_2015, title={Second-Order Γ-limit for the Cahn–Hilliard Functional}, volume={219}, DOI={10.1007/s00205-015-0924-4}, abstractNote={The goal of this paper is to solve a long standing open problem, namely, the asymptotic development of order 2 by Γ-convergence of the mass-constrained Cahn–Hilliard functional.}, number={3}, journal={Archive for Rational Mechanics and Analysis}, publisher={Springer Science and Business Media LLC}, author={Leoni, Giovanni and Murray, Ryan}, year={2015}, month={Sep}, pages={1383–1451} }