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Published December 19, 2012 | Submitted + Published
Journal Article Open

# Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting

## Abstract

In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix X formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose X into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance. The third problem is a basic geometric question: given points v1, v2, … , vn ∈ R^k (where n > k) determine whether there is a centered ellipsoid passing exactly through all of the points. We show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a subspace U that ensures any positive semidefinite matrix L with column space U can be recovered from D+L for any diagonal matrix D using a convex optimization-based heuristic known as minimum trace factor analysis. This result leads to a new understanding of the structure of rank-deficient correlation matrices and a simple condition on a set of points that ensures there is a centered ellipsoid passing through them.

© 2012 Society for Industrial and Applied Mathematics. Received by the editors April 5, 2012; accepted for publication (in revised form) by M. L. Overton October 17, 2012; published electronically December 19, 2012. This research was funded in part by Shell International Exploration and Production, Inc., under P.O. 450004440 and in part by the Air Force Office of Scientific Research under grant FA9550-11-1-0305. The authors would like to thank Prof. Sanjoy Mitter for helpful discussions and the anonymous reviewers for carefully reading the manuscript and providing many helpful suggestions.

## Attached Files

Published - 120872516.pdf

Submitted - scpw_dlr_preprint12.pdf

## Files

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