Convergence analysis of the transformed gradient projection algorithms on compact matrix manifolds

TL;DR

This paper introduces a unified framework called Transformed Gradient Projection (TGP) for optimization on compact matrix manifolds, analyzing its convergence properties and demonstrating its effectiveness through theoretical and numerical results.

Contribution

The paper develops a comprehensive TGP framework that generalizes existing algorithms and provides new convergence analysis for projection-based line-search methods on matrix manifolds.

Findings

TGP algorithms converge weakly and globally under various stepsizes

Convergence results extend or improve upon existing literature for Stiefel and Grassmann manifolds

Different scaling and normal components in TGP affect practical performance

Abstract

In this paper, we study the optimization problem on a compact matrix manifold. While existing feasible algorithms can be broadly categorized into retraction-based and projection-based methods, compared to the more comprehensive and in-depth algorithmic and convergence research framework for retraction-based line-search (RetrLS) algorithms using only tangent vectors, the theoretical understanding and algorithmic design of projection-based line-search (ProjLS) algorithms remain limited, especially when general search directions and stepsizes are involved. To bridge this gap, we propose a unified algorithmic framework called the Transformed Gradient Projection (TGP) algorithm. The key idea is to construct the search direction as a transformed Riemannian (or Euclidean) gradient augmented by an additional normal component, allowing the framework to encompass and generalize numerous existing algorithms. Then, we conduct a thorough exploration of the convergence properties of the TGP algorithms under various stepsizes, including the Armijo, Zhang-Hager type nonmonotone Armijo, and fixed stepsizes. To achieve this, we extensively analyze the geometric properties of the projection onto compact matrix manifolds, which may be of independent interest. Building upon these insights, we establish the weak convergence, iteration complexity, and global convergence of TGP algorithms under three distinct stepsizes. In cases where the compact matrix manifold is the Stiefel or Grassmann manifold, our convergence results either encompass or surpass those found in the literature. Finally, through a series of numerical experiments and theoretical analysis, we observe that different choices of scaling matrices and normal components in the search direction of TGP algorithms can lead to significantly different performance in practice.