Factorization of Binary Matrices: Rank Relations, Uniqueness and Model Selection of Boolean Decomposition

TL;DR

This paper explores various matrix factorizations of binary matrices using different operations, analyzes their uniqueness, and introduces a robust method for selecting the optimal Boolean model with accurate latent feature determination.

Contribution

It characterizes conditions for uniqueness in Boolean matrix factorizations and proposes BMF$k$, a new method for robust Boolean model selection.

Findings

BMF$k$ accurately identifies the number of Boolean latent features.

The paper establishes conditions for the uniqueness of Boolean factorizations.

Numerical experiments demonstrate the effectiveness of BMF$k$ in reconstructing factors.

Abstract

The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank decomposition is often seen as an advantageous method to interpret and analyze data. In this work, we examine the factorizations of binary matrices using standard arithmetic (real and nonnegative) and logical operations (Boolean and $\mathbb{Z}_2$). We examine the relationships between the different ranks, and discuss when factorization is unique. In particular, we characterize when a Boolean factorization $X = W \land H$ has a unique $W$, a unique $H$ (for a fixed $W$), and when both $W$ and $H$ are unique, given a rank constraint. We introduce a method for robust Boolean model selection, called BMF$k$, and show on numerical examples that BMF$k$ not only accurately determines the correct number of Boolean latent features but reconstruct the pre-determined factors accurately.