Singular value soft-thresholding via the polar decomposition

2026-07-24

AI Summary

This work addresses the computational inefficiency of traditional singular value soft-thresholding (SVT) on GPUs, which stems from its reliance on explicit singular value decomposition (SVD). The authors propose a novel approach that reformulates the SVT problem as a matrix polar decomposition and leverages an efficient, GPU-optimized polar decomposition algorithm to accelerate computation. By circumventing explicit SVD while preserving numerical stability, the method achieves substantial speedups. Experimental results demonstrate significant performance gains over standard SVD-based implementations on GPU platforms, offering an efficient computational tool for large-scale low-rank optimization tasks.

Abstract

Singular value soft-thresholding can be computed via a reduction to the matrix polar decomposition, which allows one to exploit GPU-friendly algorithms for computing the polar decomposition. Empirically, there is a significant speed-up on GPUs compared to the standard approach using the SVD. We leave the investigation of robustness to future work, but note that due to the discontinuous nature of the sign function, the reduction to the polar decomposition is likely only suitable for low-accuracy applications.

Problem

• singular value soft-thresholding
• polar decomposition
• GPU acceleration
• matrix computation
• computational efficiency

Innovation

• singular value soft-thresholding
• polar decomposition
• GPU acceleration
• matrix algorithms
• computational efficiency