# Generalized Singular Value Thresholding

arXiv: https://arxiv.org/abs/1412.2231 · HTML: https://mlsanity.com/paper/1412.2231/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Dec 2014.

## Built on by (most cited first, 8 of 8)

- [Low-Rank Tensor Completion via Novel Sparsity-Inducing Regularizers](https://mlsanity.com/paper/2310.06233/) (2023)
- [Tensor Full Feature Measure and Its Nonconvex Relaxation Applications to Tensor Recovery](https://mlsanity.com/paper/2109.12257/) (2021)
- [Provable Low Rank Plus Sparse Matrix Separation Via Nonconvex Regularizers](https://mlsanity.com/paper/2109.12713/) (2021)
- [Low-Rank Tensor Completion Based on Bivariate Equivalent Minimax-Concave Penalty](https://mlsanity.com/paper/2201.12709/) (2022)
- [Robust Low-Rank Matrix Completion via a New Sparsity-Inducing Regularizer](https://mlsanity.com/paper/2310.04762/) (2023)
- [A framework to generate sparsity-inducing regularizers for enhanced low-rank matrix completion](https://mlsanity.com/paper/2310.04954/) (2023)
- [Efficient Low-rank Identification via Accelerated Iteratively Reweighted Nuclear Norm Minimization](https://mlsanity.com/paper/2406.15713/) (2024)
- [Tailed Low-Rank Matrix Factorization for Similarity Matrix Completion](https://mlsanity.com/paper/2409.19550/) (2024)

## Abstract (arXiv)

This work studies the Generalized Singular Value Thresholding (GSVT) operator Proxg^σ(·), beginequation* Proxg^σ(B)=argminlimitsXsumi=1mg(σi(X)) + frac12||X-B||F2, endequation* associated with a nonconvex function g defined on the singular values of X. We prove that GSVT can be obtained by performing the proximal operator of g (denoted as Proxg(·)) on the singular values since Proxg(·) is monotone when g is lower bounded. If the nonconvex g satisfies some conditions (many popular nonconvex surrogate functions, e.g., ℓp-norm, 0<p<1, of ℓ0-norm are special cases), a general solver to find Proxg(b) is proposed for any b≥0. GSVT greatly generalizes the known Singular Value Thresholding (SVT) which is a basic subroutine in many convex low rank minimization methods. We are able to solve the nonconvex low rank minimization problem by using GSVT in place of SVT.