A Bayesian Boolean Matrix Factorization with Application to Copy Number Analysis in Cancer

2026-06-16

AI Summary

Existing Boolean matrix factorization methods are predominantly heuristic, sensitive to initialization, prone to local optima, and lack capabilities for model selection and uncertainty quantification, hindering their ability to uncover discrete co-variation patterns in cancer genomics. This work proposes Bayesian Boolean Matrix Factorization (BBMF), the first approach to integrate a fully conjugate Bayesian generative model into this domain. BBMF enforces strict Boolean constraints via logical AND/OR operations and leverages sparsity-inducing priors with Gibbs sampling for efficient posterior inference. The method enables automatic model selection and principled uncertainty quantification. Applied to chromosome arm-level copy number variation data from multiple myeloma patients, BBMF successfully identifies interpretable biclusters that precisely associate patient subgroups with co-varying chromosomal arms, substantially enhancing the interpretability and robustness of the inferred factors.

Abstract

Binary data factorization is common, but real-valued methods ignore discreteness and yield hard-to-interpret factors. Boolean Matrix Factorization (BooMF) instead decomposes a binary matrix into two lower-rank binary matrices via logical AND and OR, expressing the data as a Boolean disjunction of interpretable patterns. In cancer genomics, BooMF can reveal coordinated feature changes that may drive tumor evolution, unlike rotational or additive decompositions. Most existing BooMF methods are heuristic, greedy, sensitive to initialization, prone to local optima, and do not support principled model selection or uncertainty quantification. We introduce Bayesian Boolean Matrix Factorization (BBMF), a fully conjugate generative model with sparsity-inducing priors. It enforces Boolean constraints, yields interpretable latent factors with coherent uncertainty quantification, and admits Gibbs sampling with closed-form full conditionals. Because cancer evolution often involves widespread, near-simultaneous chromosome-number changes (e.g., whole-genome duplication followed by instability and selection), Boolean factorizations capture these patterns more naturally than additive models. Applied to arm-level copy-number alteration data in multiple myeloma, where entries indicate presence/absence of chromosomal-arm amplifications, BBMF finds a small set of interpretable bicliques linking patient subsets to recurrently co-altered chromosomal arms, providing a compact, biologically meaningful summary of tumor heterogeneity and demonstrating BBMF's utility for uncovering discrete latent structure in complex binary data.

Problem

• Boolean Matrix Factorization
• binary data
• cancer genomics
• copy number alteration
• latent structure

Innovation

• Bayesian Boolean Matrix Factorization
• Boolean matrix factorization
• sparsity-inducing priors
• uncertainty quantification
• cancer genomics