Binary Component Decomposition Part I: The Positive-Semidefinite Case

Abstract: This paper studies the problem of decomposing a low-rank positive-semidefinite matrix into symmetric factors with binary entries, either ${\pm 1}$ or ${0,1}$. This research answers fundamental questions about the existence and uniqueness of these decompositions. It also leads to tractable factorization algorithms that succeed under a mild deterministic condition. A companion paper addresses the related problem of decomposing a low-rank rectangular matrix into a binary factor and an unconstrained factor.