# Binary Component Decomposition Part I: The Positive-Semidefinite Case

## TL;DR

This paper investigates the decomposition of low-rank positive-semidefinite matrices into binary symmetric factors, establishing existence, uniqueness, and providing algorithms under mild conditions.

## Contribution

It introduces fundamental results on existence and uniqueness, along with tractable algorithms for binary decomposition of positive-semidefinite matrices.

## Key findings

- Proves conditions for existence of binary decompositions.
- Establishes uniqueness of such decompositions.
- Develops algorithms that succeed under mild deterministic conditions.

## 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.