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Matrix Nearness Problems with Bregman Divergences

Dhillon, Inderjit S. and Tropp, Joel A. (2007) Matrix Nearness Problems with Bregman Divergences. SIAM Journal on Matrix Analysis and Applications, 29 (4). pp. 1120-1146. ISSN 0895-4798. https://resolver.caltech.edu/CaltechAUTHORS:DHIsiamjmaa07

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Abstract

This paper discusses a new class of matrix nearness problems that measure approximation error using a directed distance measure called a Bregman divergence. Bregman divergences offer an important generalization of the squared Frobenius norm and relative entropy, and they all share fundamental geometric properties. In addition, these divergences are intimately connected with exponential families of probability distributions. Therefore, it is natural to study matrix approximation problems with respect to Bregman divergences. This article proposes a framework for studying these problems, discusses some specific matrix nearness problems, and provides algorithms for solving them numerically. These algorithms apply to many classical and novel problems, and they admit a striking geometric interpretation.


Item Type:Article
ORCID:
AuthorORCID
Tropp, Joel A.0000-0003-1024-1791
Additional Information:©2007 Society for Industrial and Applied Mathematics. Reprinted with permission. (Received January 4, 2006; accepted March 20, 2007; published November 21, 2007) This author’s [I.S.D.] research was supported by NSF grant CCF-0431257, NSF career award ACI-0093404, and NSF-ITR award IIS-0325116. This author’s [J.A.T.] research was supported by an NSF graduate fellowship. We would like to thank Nick Higham and two anonymous referees for a thorough reading and helpful suggestions.
Subject Keywords:matrix nearness problems; Bregman divergences; squared Euclidean distance; relative entropy; alternating projections
Issue or Number:4
Record Number:CaltechAUTHORS:DHIsiamjmaa07
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:DHIsiamjmaa07
Alternative URL:http://dx.doi.org/10.1137/060649021
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9428
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:30 Dec 2007
Last Modified:03 Oct 2019 00:00

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