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A Quasi-Likelihood Approach to Nonnegative Matrix Factorization.

Karthik Devarajan1, Vincent C K Cheung2

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This study introduces a unified framework for nonnegative matrix factorization using generalized linear models. The approach handles signal-dependent noise and models nonlinear effects, with applications in biomedical signal processing.

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Area of Science:

  • Statistics
  • Machine Learning
  • Biomedical Signal Processing

Background:

  • Nonnegative matrix factorization (NMF) is a widely used dimensionality reduction technique.
  • Existing NMF methods often lack a unified theoretical foundation and struggle with signal-dependent noise.
  • Statistical models offer powerful tools for data analysis but are not always integrated with NMF.

Purpose of the Study:

  • To propose a unified approach to nonnegative matrix factorization (NMF) grounded in generalized linear models (GLMs).
  • To develop a flexible framework accommodating various statistical models and enabling the modeling of nonlinear effects.
  • To introduce algorithms for NMF that effectively handle signal-dependent noise and assess factorization quality.

Main Methods:

  • Developed a unified NMF framework based on generalized linear models and quasi-likelihood theory.
  • Incorporated statistical models, including the exponential family, into the NMF framework.
  • Utilized the expectation-maximization (EM) algorithm to prove convergence of novel algorithms for signal-dependent noise.
  • Introduced a goodness-of-fit measure for evaluating NMF results.

Main Results:

  • Demonstrated that the proposed framework unifies various NMF approaches under a single theoretical umbrella.
  • Developed and proved convergence of EM-based algorithms for NMF with signal-dependent noise.
  • Showcased the ability to model nonlinear effects using link functions within the NMF framework.
  • Successfully applied the methods to biomedical signal processing tasks.

Conclusions:

  • The proposed unified NMF approach based on GLMs provides a flexible and powerful tool for data analysis.
  • The developed algorithms effectively address signal-dependent noise and allow for modeling of complex data structures.
  • This framework offers a new perspective on NMF, integrating statistical modeling principles for enhanced performance and interpretability, particularly in biomedical applications.