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

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 ∈ Rk (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 optimizationbased 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.