A unified framework for sparse non-negative least squares using multiplicative updates and the non-negative matrix...

Area of Science:

• Optimization algorithms
• Machine learning
• Signal processing

Background:

• Sparse non-negative least squares (S-NNLS) is crucial for recovering non-negative quantities from linear measurements in various applications.
• Existing S-NNLS algorithms lack a unified theoretical framework, hindering broader applicability and algorithm development.

Purpose of the Study:

• To present a unified framework for S-NNLS based on a rectified power exponential scale mixture prior.
• To develop computationally efficient inference procedures for S-NNLS and its extension to sparse non-negative matrix factorization (S-NMF).

Main Methods:

• A novel unified framework for S-NNLS utilizing a rectified power exponential scale mixture prior on sparse codes.
• Development of multiplicative update rules for efficient computation and guaranteed convergence to stationary points.
• Extension of the framework to S-NMF, unifying several existing algorithms.

Main Results:

• The proposed framework encompasses a wide range of S-NNLS algorithms.
• Multiplicative update rules provide efficient solutions for large-scale S-NNLS and S-NMF problems.
• Theoretical guarantees for convergence to stationary points and sparsity of local minima are established.

Conclusions:

• The unified framework offers a principled approach to S-NNLS and S-NMF, enhancing algorithmic development and theoretical understanding.
• The efficient update rules facilitate practical application in data recovery and factorization tasks.
• Empirical validation on synthetic and real-world data demonstrates the effectiveness of the proposed methods.