Low-Rank Positive Approximants of Symmetric Matrices

Abstract

Given a symmetric matrix X , we consider the problem of finding a low-rank positive approximant of X . That is, a symmetric positive semidefinite matrix, S , whose rank is smaller than a given positive integer, , which is nearest to X in a certain matrix norm. The problem is first solved with regard to four common norms: The Frobenius norm, the Schatten p -norm, the trace norm, and the spectral norm. Then the solution is extended to any unitarily invariant matrix norm. The proof is based on a subtle combination of Ky Fan dominance theorem, a modified pinching principle, and Mirsky minimum-norm theorem.