Generalized Unitarily Invariant Gauge Regularization for Fast Low-Rank Matrix Recovery

Area of Science:

• Numerical Analysis
• Linear Algebra
• Machine Learning

Background:

• Low-rank matrix recovery (LRMR) commonly uses spectral regularization, which relies on singular value decomposition (SVD).
• SVD computation is a bottleneck for large-scale LRMR, limiting practical applications.
• Existing methods struggle with efficiency and scalability due to iterative SVD calculations.

Purpose of the Study:

• To develop a novel regularization approach for LRMR that bypasses the need for SVD.
• To generalize existing spectral functions within a new framework.
• To enable efficient and accurate LRMR for large-scale datasets.

Main Methods:

• Introduction of a generalized unitarily invariant gauge (GUIG) function for LRMR.
• Formulation of the GUIG regularization model as a bilinear variational problem.
• Development of an efficient algorithm that avoids explicit SVD computation.

Main Results:

• The GUIG function generalizes spectral functions like the rank, Schatten-p norm, and logsum of singular values.
• The proposed method achieves efficient solutions without SVD, suitable for large-scale LRMR.
• Experimental validation on matrix completion and robust principal component analysis demonstrates superior accuracy and speed.

Conclusions:

• The GUIG regularization offers a computationally efficient and accurate alternative to traditional spectral methods for LRMR.
• The SVD-free approach significantly enhances the scalability of LRMR algorithms.
• The GUIG method shows strong performance, outperforming state-of-the-art algorithms on large-scale problems.