Fast Optimization Algorithm on Riemannian Manifolds and Its Application in Low-Rank Representation

TL;DR

This paper introduces a fast first-order optimization algorithm with quadratic convergence for Riemannian manifold problems, demonstrating superior performance in low-rank matrix completion tasks.

Contribution

It proposes a novel FOA with quadratic convergence for Riemannian optimization and applies it to low-rank matrix representation, improving convergence speed and accuracy.

Findings

FOA outperforms existing methods in matrix completion tasks.

Experimental results show faster convergence and higher accuracy.

The proposed methods are effective on synthetic and real datasets.

Abstract

The paper addresses the problem of optimizing a class of composite functions on Riemannian manifolds and a new first order optimization algorithm (FOA) with a fast convergence rate is proposed. Through the theoretical analysis for FOA, it has been proved that the algorithm has quadratic convergence. The experiments in the matrix completion task show that FOA has better performance than other first order optimization methods on Riemannian manifolds. A fast subspace pursuit method based on FOA is proposed to solve the low-rank representation model based on augmented Lagrange method on the low rank matrix variety. Experimental results on synthetic and real data sets are presented to demonstrate that both FOA and SP-RPRG(ALM) can achieve superior performance in terms of faster convergence and higher accuracy.