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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. \ud \ud \ud 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

Publisher: Society for Industrial and Applied Mathematics

Year: 2012

OAI identifier:
oai:authors.library.caltech.edu:34685

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Caltech Authors

- (2007). Algebraic factor analysis: tetrads, pentads and beyond, Probab. Theory Related Fields,
- (1998). Approximations for partially coherent optical imaging systems,
- (1993). Combinatorial properties and the complexity of a max-cut approximation,
- (2002). Computing the nearest correlation matrix—a problem from finance,
- (2011). Equivariant Gro¨bner bases and the Gaussian two-factor model,
- (2009). Exact matrix completion via convex optimization,
- (1990). Extremal correlation matrices,
- (1904). General intelligence,’ objectively determined and measured,
- (2010). Grothendieck inequalities for semidefinite programs with rank constraint, Arxiv preprint arXiv:1011.1754,
- (2011). Group symmetry and covariance regularization, Arxiv preprint arXiv:1111.7061,
- (2010). Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,
- (1985). Identifiability of factor analysis: some results and open problems,
- (1985). Identification of noisy systems,
- (1995). Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,
- (1977). Maximum likelihood from incomplete data via the EM algorithm,
- (1982). Minimum rank and minimum trace of covariance matrices,
- (1969). Nonnegative matrix equations having positive solutions,
- (1995). On a positive semidefinite relaxation of the cut polytope,
- (1940). On a problem concerning matrices with variable diagonal elements,
- (1995). On the facial structure of the set of correlation matrices,
- (1997). On the rank minimization problem over a positive semidefinite linear matrix inequality,
- (1982). Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis,
- (2011). Rank-sparsity incoherence for matrix decomposition,
- (2011). Robust principal component analysis?,
- (1996). Semidefinite programming,
- (1990). Some geometric applications of the beta distribution,
- (2011). Subspace identification via convex optimization, master’s thesis,
- (1944). The matrices of factor analysis,
- (1996). Two decades of array signal processing research,
- (2001). Uncertainty principles and ideal atomic decomposition,
- (1982). Weighted minimum trace factor analysis,