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

Desantis, Derek Freeman; Skau, Erik West; Truong, Phan Minh Duc; Alexandrov, Boian

doi:10.1145/3522594

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

Journal Article · 29 July 2022 · ACM Transactions on Knowledge Discovery from Data

DOI: https://doi.org/10.1145/3522594 · OSTI ID:2310318

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Los Alamos National Laboratory (LANL), Los Alamos, NM (United States). Center for Nonlinear Studies (CNLS)
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

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 BMFk, and show on numerical examples that BMFk not only accurately determines the correct number of Boolean latent features but reconstruct the pre-determined factors accurately.

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Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2310318

Report Number(s):: LA-UR--21-30686

Journal Information:: ACM Transactions on Knowledge Discovery from Data, Journal Name: ACM Transactions on Knowledge Discovery from Data Journal Issue: 6 Vol. 16; ISSN 1556-4681

Publisher:: Association for Computing Machinery (ACM)Copyright Statement

Country of Publication:: United States

Language:: English

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