# Bayesian Mean-parameterized Nonnegative Binary Matrix Factorization

## TL;DR

This paper introduces a Bayesian mean-parameterized nonnegative binary matrix factorization framework that avoids link functions, includes novel inference algorithms, and automatically determines the number of latent factors, improving interpretability and performance.

## Contribution

It proposes a unified Bayesian mean-parameterized NMF model for binary data, with new inference algorithms and a nonparametric extension for automatic latent dimension selection.

## Key findings

- Achieves comparable or better results than existing methods.
- Automatically detects the number of relevant components.
- Effective in tasks like dictionary learning and missing data prediction.

## Abstract

Binary data matrices can represent many types of data such as social networks, votes, or gene expression. In some cases, the analysis of binary matrices can be tackled with nonnegative matrix factorization (NMF), where the observed data matrix is approximated by the product of two smaller nonnegative matrices. In this context, probabilistic NMF assumes a generative model where the data is usually Bernoulli-distributed. Often, a link function is used to map the factorization to the $[0,1]$ range, ensuring a valid Bernoulli mean parameter. However, link functions have the potential disadvantage to lead to uninterpretable models. Mean-parameterized NMF, on the contrary, overcomes this problem. We propose a unified framework for Bayesian mean-parameterized nonnegative binary matrix factorization models (NBMF). We analyze three models which correspond to three possible constraints that respect the mean-parametrization without the need for link functions. Furthermore, we derive a novel collapsed Gibbs sampler and a collapsed variational algorithm to infer the posterior distribution of the factors. Next, we extend the proposed models to a nonparametric setting where the number of used latent dimensions is automatically driven by the observed data. We analyze the performance of our NBMF methods in multiple datasets for different tasks such as dictionary learning and prediction of missing data. Experiments show that our methods provide similar or superior results than the state of the art, while automatically detecting the number of relevant components.