Tangent Space Based Alternating Projections for Nonnegative Low Rank Matrix Approximation

TL;DR

This paper introduces a tangent space-based alternating projection method for nonnegative low rank matrix approximation, reducing computational costs and improving convergence over traditional methods.

Contribution

It proposes a novel tangent space approach to approximate projections, enabling efficient and linearly convergent solutions for nonnegative low rank matrix approximation.

Findings

Method outperforms nonnegative matrix factorization in speed and accuracy.

Converges linearly to near-optimal solutions.

Effective in data clustering, pattern recognition, hyperspectral analysis.

Abstract

In this paper, we develop a new alternating projection method to compute nonnegative low rank matrix approximation for nonnegative matrices. In the nonnegative low rank matrix approximation method, the projection onto the manifold of fixed rank matrices can be expensive as the singular value decomposition is required. 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 the alternating projections onto the tangent spaces of the fixed rank matrices manifold and the nonnegative matrix manifold, converge linearly to a point in the intersection of the two manifolds where the convergent point is sufficiently close to optimal solutions. This convergence result based inexact projection onto the manifold is new and is not studied in the literature. Numerical examples in data clustering, pattern recognition and hyperspectral data analysis are given to demonstrate that the performance of the proposed method is better than that of nonnegative matrix factorization methods in terms of computational time and accuracy.