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 \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.Comment: 20 page