Rank Restricted Semidefinite Matrices and Image Closedness

TL;DR

This paper investigates the closure properties of rank-restricted positive semidefinite matrices and their implications for low-rank matrix completion problems, providing theoretical insights into the conditions for successful completion.

Contribution

It introduces a new analysis of the closure of rank-restricted semidefinite matrices and explores its applications to low-rank matrix completion via nuclear norm heuristics.

Findings

Characterizes the closure of rank-restricted semidefinite matrices.

Provides conditions for low-rank matrix completion success.

Links closure properties to nuclear norm heuristic effectiveness.

Abstract

We study the closure of the projection of the (nonconvex) cone of rank restricted positive semidefinite matrices onto subsets of the matrix entries. This defines the feasible sets for semidefinite completion problems with restrictions on the ranks. Applications include conditions for low-rank completions using the nuclear norm heuristic.