Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting
Name
Saunderson-2012-DIAGONAL AND LOW-RANK MATRIX DECOMPOSITIONS.pdf
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Author(s)
Saunderson, James F.
Chandrasekaran, Venkat
Parrilo, Pablo A.
Willsky, Alan S.
Date Issued
December 2012
Journal
SIAM Journal on Matrix Analysis and Applications
Publisher
Society for Industrial and Applied Mathematics
Citation
Saunderson, J. et al. “Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting.” SIAM Journal on Matrix Analysis and Applications 33.4 (2012): 1395–1416.
Version
Final published version
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 $v_1,v_2,\ldots,v_n\in \mathbb{R}^k$ (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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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DOI of Published Version
https://doi.org/10.1137/120872516
