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

• Main Authors: Saunderson, James F. (Contributor), Chandrasekaran, Venkat (Author), Parrilo, Pablo A. (Contributor), Willsky, Alan S. (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor), Massachusetts Institute of Technology. Laboratory for Information and Decision Systems (Contributor)
• Format: Article
• Language: English
• Published: Society for Industrial and Applied Mathematics, 2013-03-12T18:18:29Z.
• Subjects: Article
• Online Access: Get fulltext

 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 $v_1,v_2,\ldots,v_n\in \mathbb{R}^k$ (where $n > k$) determine whether there is a centered ellipsoid passing exactly through all the points. We show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a subspace $\mathcal{U}$ that ensures any positive semidefinite matrix $L$ with column space $\mathcal{U}$ 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.