# A smoothing proximal gradient algorithm for matrix rank minimization problem

arXiv: https://arxiv.org/abs/2103.09530 · HTML: https://mlsanity.com/paper/2103.09530/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Mar 2021.

## Built on by (most cited first, 2 of 2)

- [Federated Smoothing Proximal Gradient for Quantile Regression with Non-Convex Penalties](https://mlsanity.com/paper/2408.05640/) (2024)
- [A Proximal Modified Quasi-Newton Method for Nonsmooth Regularized Optimization](https://mlsanity.com/paper/2409.19428/) (2024)

## Abstract (arXiv)

In this paper, we study the low-rank matrix minimization problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. We first introduce an exact continuous relaxation, that is, both problems have the same minimzers and the same optimal value. In particular, we introduce a class of lifted stationary point of the relaxed problem and show that any local minimizer of the relaxed problem must be a lifted stationary point. In addition, we derive lower bound property for the nonzero singular values of the lifted stationary point and hence also of the local minimizers of the relaxed problem. Then the smoothing proximal gradient (SPG) algorithm is proposed to find a lifted stationary point of the continuous relaxation model. Moreover, it is shown that the whole sequence generated by SPG algorithm converges to a lifted stationary point. At last, numerical examples show the efficiency of the SPG algorithm.