Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem.

Area of Science:

• Signal Processing
• Machine Learning
• Statistical Inference

Background:

• Sparse non-negative least squares (S-NNLS) is a fundamental problem in signal processing.
• Existing S-NNLS solvers often struggle with signal and support recovery accuracy.
• Bayesian methods offer a principled approach to incorporate prior information, enhancing sparsity assumptions.

Purpose of the Study:

• To develop a novel Bayesian evidence maximization framework for S-NNLS.
• To introduce the Rectified Gaussian Scale Mixture (R-GSM) prior for enforcing sparsity.
• To propose a new method, rectified Sparse Bayesian Learning (R-SBL), for robust S-NNLS solutions.

Main Methods:

• Developed a Bayesian evidence maximization framework using the Expectation-Maximization (EM) algorithm.
• Introduced the Rectified Gaussian Scale Mixture (R-GSM) prior, encompassing heavy-tailed distributions.
• Proposed four EM-based R-SBL variants trading off computational complexity and accuracy (MCMC EM, MMSE, AMP, diagonal approximation).

Main Results:

• The proposed R-SBL method demonstrated superior performance in both signal and support recovery compared to existing S-NNLS solvers.
• Numerical experiments confirmed the robustness of R-SBL against various design matrix structures.
• The R-GSM prior effectively models sparsity through distributions like rectified Laplacian and Student-t.

Conclusions:

• The R-SBL framework provides an effective and robust solution for the sparse non-negative least squares problem.
• The R-GSM prior offers flexibility in modeling sparsity, leading to improved recovery performance.
• The proposed EM-based variants offer practical options for applying R-SBL in diverse computational scenarios.