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

2026-02-27

AI Summary

This work addresses the recovery of low-rank solutions to linear inverse problems by studying Schatten-$p$ quasi-norm regularization for $p \in [0,1]$. A novel dynamic proximal gradient algorithm is proposed, which—by incorporating the Cayley transform for the first time in this context—effectively circumvents the costly singular value decomposition at each iteration. The method integrates an adaptive backtracking scheme with an explicit stepsize strategy and is analyzed within the Kurdyka–Łojasiewicz (KL) framework. Global sequential convergence and a sublinear convergence rate are established for any $p \in [0,1]$; furthermore, under standard regularity conditions, linear convergence is achieved when $p = 1$. Numerical experiments demonstrate the algorithm’s significant computational efficiency over existing approaches.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

This paper investigates numerical solution methods for the Schatten-$p$ quasi-norm regularized problem with $p \in [0,1]$, 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 $p \in [0,1]$ 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.

Problem

• Schatten-p quasi-norm
• low-rank recovery
• linear inverse problems
• nonconvex optimization

Innovation

• Schatten-p quasi-norm
• proximal gradient algorithm
• Cayley transformation
• singular value decomposition avoidance
• Kurdyka-Łojasiewicz property