Variational Bayesian Orthogonal Nonnegative Matrix Factorization Over the Stiefel Manifold.

Area of Science:

• Multivariate Data Analysis
• Machine Learning
• Probabilistic Modeling

Background:

• Nonnegative Matrix Factorization (NMF) is a key multivariate data analysis technique.
• Uniqueness and rank selection remain open challenges in NMF.
• Orthogonal NMF (ONMF) offers sparser representations but often lacks probabilistic frameworks for real-world data noise and uncertainties.

Purpose of the Study:

• To propose a novel probabilistic formulation of orthogonal Nonnegative Matrix Factorization (ONMF).
• To address NMF uniqueness and model order selection issues using probabilistic methods.
• To develop an efficient inference algorithm for the proposed model.

Main Methods:

• Introduced a probabilistic ONMF model using a von Mises-Fisher distribution on the Stiefel manifold for orthogonality.
• Incorporated an Automatic Relevance Determination (ARD) prior for model order selection.
• Developed an efficient variational Bayesian inference algorithm for model solution.

Main Results:

• The proposed Variational Bayesian Orthogonal Nonnegative Matrix Factorization (VBONMF) model effectively handles noise and variable uncertainties.
• The model demonstrated efficiency and competitiveness in blind decomposition of multispectral images of ancient documents.
• Numerical experiments validated the model's performance against state-of-the-art approaches.

Conclusions:

• The developed probabilistic ONMF framework provides a robust solution to NMF challenges.
• VBONMF offers improved performance and efficiency for complex data analysis tasks.
• The model shows promise for applications involving real-world data, such as multispectral image analysis.