Convergence analysis for a variant of manifold proximal point algorithm based on Kurdyka-{\L}ojasiewicz property

TL;DR

This paper analyzes the convergence of a manifold proximal point algorithm with an reweighted strategy for sparse vector identification, establishing global convergence and linear rates under Kurdyka-Łojasiewicz properties.

Contribution

It introduces an enhanced manifold proximal point algorithm with reweighting, proving its global convergence and linear convergence rates under KL conditions.

Findings

The algorithm converges globally under KL properties.

Linear convergence is achieved when the model's optimal value is positive.

Finite convergence occurs at weak sharp minima.

Abstract

We incorporate an iteratively reweighted strategy in the manifold proximal point algorithm (ManPPA) in [12] to solve an enhanced sparsity inducing model for identifying sparse yet nonzero vectors in a given subspace. We establish the global convergence of the whole sequence generated by our algorithm by assuming the Kurdyka-Lojasiewicz (KL) properties of suitable potential functions. We also study how the KL exponents of the different potential functions are related. More importantly, when our enhanced model and algorithm reduce, respectively, to the model and ManPPA with constant stepsize considered in [12], we show that the sequence generated converges linearly as long as the optimal value of the model is positive, and converges finitely when the limit of the sequence lies in a set of weak sharp minima. Our results improve [13, Theorem 2.4], which asserts local quadratic convergence in the presence of weak sharp minima when the constant stepsize is small.