# Dynamic Proximal Gradient Algorithms for Schatten-p Quasi-Norm Regularized Problems

arXiv: https://arxiv.org/abs/2603.00333 · HTML: https://mlsanity.com/paper/2603.00333/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 66 references that are in the map.
- First posted on arXiv 27 Feb 2026.

## Builds on (most cited first, 13 of 13)

- [Error bounds, quadratic growth, and linear convergence of proximal methods](https://mlsanity.com/paper/1602.06661/) (2016)
- [Calculus of the exponent of Kurdyka-Lojasiewicz inequality and its applications to linear convergence of first-order methods](https://mlsanity.com/paper/1602.02915/) (2016)
- [Weighted Schatten p-Norm Minimization for Image Denoising and Background Subtraction](https://mlsanity.com/paper/1512.01003/) (2015)
- [Splitting methods with variable metric for KL functions](https://mlsanity.com/paper/1405.1357/) (2014)
- [A Unified Approach to Error Bounds for Structured Convex Optimization Problems](https://mlsanity.com/paper/1512.03518/) (2015)
- [Global convergence of splitting methods for nonconvex composite optimization](https://mlsanity.com/paper/1407.0753/) (2014)
- [Douglas-Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems](https://mlsanity.com/paper/1409.8444/) (2014)
- [Factor Group-Sparse Regularization for Efficient Low-Rank Matrix Recovery](https://mlsanity.com/paper/1911.05774/) (2019)
- [Local Linear Convergence of Forward-Backward under Partial Smoothness](https://mlsanity.com/paper/1407.5611/) (2014)
- [Learning Sparse and Low-Rank Priors for Image Recovery via Iterative Reweighted Least Squares Minimization](https://mlsanity.com/paper/2304.10536/) (2023)
- [Rotation Group Synchronization via Quotient Manifold](https://mlsanity.com/paper/2306.12730/) (2023)
- [Scalable Algorithms for Tractable Schatten Quasi-Norm Minimization](https://mlsanity.com/paper/1606.01245/) (2016)
- [Fast Singular Value Shrinkage with Chebyshev Polynomial Approximation Based on Signal Sparsity](https://mlsanity.com/paper/1705.07112/) (2017)

## Abstract (arXiv)

This paper investigates numerical solution methods for the Schatten-p quasi-norm regularized problem with p ∈ [0,1], which has been widely studied for finding low-rank solutions of linear inverse problems and gained successful applications in various mathematics and applied science fields. We propose a dynamic proximal gradient algorithm that, through the use of the Cayley transformation, avoids computationally expensive singular value decompositions at each iteration, thereby significantly reducing the computational complexity. The algorithm incorporates two step size selection strategies: an adaptive backtracking search and an explicit step size rule. We establish the sublinear convergence of the proposed algorithm for all p ∈ [0,1] within the framework of the Kurdyka-Lojasiewicz property. Notably, under mild assumptions, we show that the generated sequence converges to a stationary point of the objective function of the problem. For the special case when p=1, the linear convergence is further proved under the strict complementarity-type regularity condition commonly used in the linear convergence analysis of the forward-backward splitting algorithms. Preliminary numerical results validate the superior computational efficiency of the proposed algorithm.