# Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm

arXiv: https://arxiv.org/abs/1510.06895 · HTML: https://mlsanity.com/paper/1510.06895/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2015.

## Built on by (most cited first, 23 of 23)

- [GaitGCI: Generative Counterfactual Intervention for Gait Recognition](https://mlsanity.com/paper/2306.03428/) (2023)
- [A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples](https://mlsanity.com/paper/2106.02119/) (2021)
- [A Framework of Inertial Alternating Direction Method of Multipliers for Non-Convex Non-Smooth Optimization](https://mlsanity.com/paper/2102.05433/) (2021)
- [Low-Rank Tensor Completion via Novel Sparsity-Inducing Regularizers](https://mlsanity.com/paper/2310.06233/) (2023)
- [Deep Unfolding of Iteratively Reweighted ADMM for Wireless RF Sensing](https://mlsanity.com/paper/2106.03686/) (2021)
- [Multi-Tensor Network Representation for High-Order Tensor Completion](https://mlsanity.com/paper/2109.04022/) (2021)
- [Low-rank Tensor Learning with Nonconvex Overlapped Nuclear Norm Regularization](https://mlsanity.com/paper/2205.03059/) (2022)
- [Learning Transition Operators From Sparse Space-Time Samples](https://mlsanity.com/paper/2212.00746/) (2022)
- [Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion](https://mlsanity.com/paper/2504.21468/) (2025)
- [Efficient Differentiable Approximation of Generalized Low-rank Regularization](https://mlsanity.com/paper/2505.15407/) (2025)
- [Provable Low Rank Plus Sparse Matrix Separation Via Nonconvex Regularizers](https://mlsanity.com/paper/2109.12713/) (2021)
- ["Sparse + Low-Rank'' Tensor Completion Approach for Recovering Images and Videos](https://mlsanity.com/paper/2110.09298/) (2021)
- [Multiblock ADMM for nonsmooth nonconvex optimization with nonlinear coupling constraints](https://mlsanity.com/paper/2201.07657/) (2022)
- [Introducing the Huber mechanism for differentially private low-rank matrix completion](https://mlsanity.com/paper/2206.07910/) (2022)
- [Region-wise matching for image inpainting based on adaptive weighted low-rank decomposition](https://mlsanity.com/paper/2303.12421/) (2023)
- [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)
- [Tailed Low-Rank Matrix Factorization for Similarity Matrix Completion](https://mlsanity.com/paper/2409.19550/) (2024)
- [Robust Orthogonal NMF with Label Propagation for Image Clustering](https://mlsanity.com/paper/2504.21472/) (2025)
- [Accelerating Large-Scale Regularized High-Order Tensor Recovery](https://mlsanity.com/paper/2506.09594/) (2025)
- [Association and Consolidation: Evolutionary Memory-Enhanced Incremental Multi-View Clustering](https://mlsanity.com/paper/2509.14544/) (2025)
- [Simplifying Optimal Transport through Schatten-p Regularization](https://mlsanity.com/paper/2510.11910/) (2025)
- [Robust Tensor Regression with Nonconvexity: Algorithmic and Statistical Theory](https://mlsanity.com/paper/2605.07448/) (2026)

## Abstract (arXiv)

The nuclear norm is widely used as a convex surrogate of the rank function in compressive sensing for low rank matrix recovery with its applications in image recovery and signal processing. However, solving the nuclear norm based relaxed convex problem usually leads to a suboptimal solution of the original rank minimization problem. In this paper, we propose to perform a family of nonconvex surrogates of L0-norm on the singular values of a matrix to approximate the rank function. This leads to a nonconvex nonsmooth minimization problem. Then we propose to solve the problem by Iteratively Reweighted Nuclear Norm (IRNN) algorithm. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem, which has a closed form solution due to the special properties of the nonconvex surrogate functions. We also extend IRNN to solve the nonconvex problem with two or more blocks of variables. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthesized data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms.