Matrix Nearness Problems with Bregman Divergences
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.
©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.