TITLE:
Low-Rank Positive Approximants of Symmetric Matrices
AUTHORS:
Achiya Dax
KEYWORDS:
Low-Rank Positive Approximants, Unitarily Invariant Matrix Norms
JOURNAL NAME:
Advances in Linear Algebra & Matrix Theory,
Vol.4 No.3,
September
25,
2014
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.