@article{chu_funderlic_plemmons_2003, title={Structured low rank approximation}, volume={366}, ISSN={["0024-3795"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0037409774&partnerID=MN8TOARS}, DOI={10.1016/S0024-3795(02)00505-0}, abstractNote={This paper concerns the construction of a structured low rank matrix that is nearest to a given matrix. The notion of structured low rank approximation arises in various applications, ranging from signal enhancement to protein folding to computer algebra, where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system. The task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix. This paper addresses some of the theoretical and numerical issues involved in the problem. Two procedures for constructing the nearest structured low rank matrix are proposed. The procedures are flexible enough that they can be applied to any lower rank, any linear structure, and any matrix norm in the measurement of nearness. The techniques can also be easily implemented by utilizing available optimization packages. The special case of symmetric Toeplitz structure using the Frobenius matrix norm is used to exemplify the ideas throughout the discussion. The concept, rather than the implementation details, is the main emphasis of the paper.}, number={SPEC. ISS.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Chu, MT and Funderlic, RE and Plemmons, RJ}, year={2003}, month={Jun}, pages={157–172} }
 @article{chu_fundelic_2002, title={The centroid decomposition: Relationships between discrete variational decompositions and SVDs}, volume={23}, ISSN={["1095-7162"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0036401834&partnerID=MN8TOARS}, DOI={10.1137/S0895479800382555}, abstractNote={The centroid decomposition, an approximation for the singular value decomposition (SVD), has a long history among the statistics/psychometrics community for factor analysis research. We revisit the centroid method in its original context of factor analysis and then adapt it to other than a covariance matrix. The centroid method can be cast as an ${\cal O}(n)$-step ascent method on a hypercube. It is shown empirically that the centroid decomposition provides a measurement of second order statistical information of the original data in the direction of the corresponding left centroid vectors. One major purpose of this work is to show fundamental relationships between the singular value, centroid, and semidiscrete decompositions. This unifies an entire class of truncated SVD approximations. Applications include semantic indexing in information retrieval.}, number={4}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, MT and Fundelic, RE}, year={2002}, month={May}, pages={1025–1044} }
 @article{chu_funderlic_golub_1998, title={Rank modifications of semidefinite matrices associated with a secant update formula}, volume={20}, ISSN={["0895-4798"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0032216893&partnerID=MN8TOARS}, DOI={10.1137/S0895479896306021}, abstractNote={This paper analyzes rank modification of symmetric positive definite matrices H of the form H −M + P , where H −M denotes a step of reducing H to a lower-rank, symmetric and positive semidefinite matrix and (H −M) + P denotes a step of restoring H −M to a symmetric positive definite matrix. These steps and their generalizations for rectangular matrices are fully characterized. The well-known BFGS and DFP updates used in Hessian and inverse Hessian approximations provided the motivation for the generalizations and are special cases with H and P having rank one.}, number={2}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, MT and Funderlic, RE and Golub, GH}, year={1998}, month={Oct}, pages={428–436} }
 @article{chu_funderlic_golub_1997, title={On a variational formulation of the generalized singular value decomposition}, volume={18}, ISSN={["0895-4798"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0031520186&partnerID=MN8TOARS}, DOI={10.1137/S0895479895287079}, abstractNote={A variational formulation for the generalized singular value decomposition (GSVD) of a pair of matrices $A \in R^{m \times n}$ and $B \in R^{p \times n}$ is presented. In particular, a duality theory analogous to that of the SVD provides new understanding of left and right generalized singular vectors. It is shown that the intersection of row spaces of $A$ and $B$ plays a key role in the GSVD duality theory. The main result that characterizes left GSVD vectors involves a generalized singular value deflation process.}, number={4}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, MT and Funderlic, RE and Golub, GH}, year={1997}, month={Oct}, pages={1082–1092} }