Fast Alternating Projections on Manifolds Based on Tangent Spaces

TL;DR

This paper introduces a tangent space-based approximation method for alternating projections on manifolds, significantly reducing computational costs while maintaining convergence to intersection points, demonstrated through matrix approximation examples.

Contribution

It proposes a novel tangent space-based approach for alternating projections on manifolds, improving efficiency over classical methods.

Findings

Converges linearly to intersection points.

Reduces computational time compared to classical methods.

Effective in low-rank matrix and image quaternion matrix approximations.

Abstract

In this paper, we study alternating projections on nontangential manifolds based on the tangent spaces. The main motivation is that the projection of a point onto a manifold can be computational expensive. We propose to use the tangent space of the point in the manifold to approximate the projection onto the manifold in order to reduce the computational cost. We show that the sequence generated by alternating projections on two nontangential manifolds based on tangent spaces, converges linearly to a point in the intersection of the two manifolds where the convergent point is close to the optimal solution. Numerical examples for nonnegative low rank matrix approximation and low rank image quaternion matrix (color image) approximation, are given to demonstrate that the performance of the proposed method is better than that of the classical alternating projection method in terms of computational time.