Bayesian Ternary Matrix Factorization for Sign‑Aware Discrete Data

Overview

This project introduces Bayesian Ternary Matrix Factorization (BTMF) for structured data whose entries take values in ({-1, 0, +1}). Such data arise naturally in genomics and other applications, for example:

• copy‑number alterations encoded as deletion (−1), normal (0), amplification (+1),
• directional effects (negative, neutral, positive), or
• signed interaction patterns.

The goal is to uncover sign‑aware latent factors that explain the observed ternary structure in a compact and interpretable way.

Model Structure

BTMF decomposes a ternary observation matrix (X) into:

• a binary activation matrix (W) – indicates which latent factors are active for each sample, and
• a ternary loadings matrix (H) – encodes negative (−1), neutral (0), or positive (+1) contributions of each factor to each feature.

The product of (W) and (H) (under an appropriate discrete generative model) yields a low‑rank, sign‑aware representation of the data. This factorization:

• separates which factors are used (driven by (W)) from
• how they affect each feature (captured in the signs of (H)).

Bayesian Priors and Inference

To ensure interpretability and robust regularization, BTMF uses:

• Beta–Bernoulli priors on the columns of (W), encouraging sparse and selective factor usage;
• multinomial–Dirichlet priors on the columns of (H), allowing the model to learn asymmetric sign patterns across (-1, 0, +1).

These conjugate priors lead to closed‑form Gibbs sampling updates, making inference efficient and scalable to larger datasets while still providing full Bayesian uncertainty quantification.

Outcomes

The Bayesian TMF framework:

• discovers coherent, low‑dimensional structure in ternary datasets,
• preserves and interprets the sign information that is often lost in purely binary models, and
• provides a principled probabilistic model suitable for copy‑number alteration data and other sign‑coded measurements.

This project complements the Boolean factorization work by extending discrete matrix factorization to richer, sign‑aware settings, broadening the range of biological and applied problems that can be addressed.