Variational Bayesian Matrix Factorization for Bounded Support Data

Area of Science:

• Computational Statistics
• Machine Learning
• Data Mining

Background:

• Matrix factorization is crucial for dimensionality reduction and pattern discovery in data analysis.
• Existing methods often struggle with bounded support data, limiting their applicability.
• Bayesian approaches offer a principled way to incorporate prior knowledge and quantify uncertainty.

Purpose of the Study:

• To introduce a novel Bayesian matrix factorization method, beta-gamma nonnegative matrix factorization (BG-NMF), specifically designed for bounded support data.
• To develop an analytically tractable method for approximating posterior distributions within the BG-NMF framework.
• To enhance the model by incorporating sparsity constraints for improved feature extraction.

Main Methods:

• The proposed BG-NMF method models bounded data using a beta distribution for the observation matrix.
• It integrates nonnegative matrix factorization (NMF) with gamma priors on the factorized matrices (basis and excitation).
• Variational inference is employed to derive an analytically tractable lower-bound for approximating posterior distributions, maintaining conjugacy.

Main Results:

• The BG-NMF model successfully approximates posterior distributions using a derived lower-bound, yielding gamma-distributed posteriors.
• Incorporating a sparseness constraint on the gamma prior enables sparse BG-NMF, enhancing interpretability.
• Evaluations on synthetic and real-world datasets confirm the effectiveness and good performance of the proposed BG-NMF method.

Conclusions:

• The BG-NMF method offers a robust Bayesian framework for matrix factorization of bounded support data.
• The developed variational inference approach provides an efficient and analytically tractable solution for posterior estimation.
• The method's flexibility, including the ability to induce sparsity, makes it a valuable tool for various data analysis applications.