%0 Generic %1 saunderson2012diagonal %A Saunderson, James %A Chandrasekaran, Venkat %A Parrilo, Pablo A. %A Willsky, Alan S. %D 2012 %K stats %R 10.1137/120872516 %T Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting %U http://arxiv.org/abs/1204.1220 %X 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 $v_1,v_2,...,v_n\R^k$ (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.

@misc{saunderson2012diagonal, 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 $v_1,v_2,...,v_n\in \R^k$ (where $n > k$) determine whether there is a centered ellipsoid passing \emph{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.}, added-at = {2013-04-03T09:27:40.000+0200}, author = {Saunderson, James and Chandrasekaran, Venkat and Parrilo, Pablo A. and Willsky, Alan S.}, biburl = {https://www.bibsonomy.org/bibtex/2b712fdaf59a1e056ba53f347f9dcedd3/jf.cardoso}, description = {Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting}, doi = {10.1137/120872516}, interhash = {c0317c2ca2876fc8eeab4f5954408bfa}, intrahash = {b712fdaf59a1e056ba53f347f9dcedd3}, keywords = {stats}, note = {cite arxiv:1204.1220Comment: 20 pages}, timestamp = {2013-04-03T09:27:40.000+0200}, title = {Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting}, url = {http://arxiv.org/abs/1204.1220}, year = 2012 }