Robustness Analysis of Preconditioned Successive Projection Algorithm for General Form of Separable NMF Problem

TL;DR

This paper demonstrates that the preconditioned Successive Projection Algorithm (SPA) remains robust to noise in separable NMF problems even when the data dimension exceeds the factorization rank, extending previous results.

Contribution

It proves the noise robustness of preconditioned SPA without requiring the condition that data dimension equals the factorization rank.

Findings

Preconditioned SPA is robust to noise when d > r.

The robustness proof does not rely on the d=r condition.

Extends applicability of SPA in real-world scenarios.

Abstract

The successive projection algorithm (SPA) has been known to work well for separable nonnegative matrix factorization (NMF) problems arising in applications, such as topic extraction from documents and endmember detection in hyperspectral images. One of the reasons is in that the algorithm is robust to noise. Gillis and Vavasis showed in [SIAM J. Optim., 25(1), pp. 677-698, 2015] that a preconditioner can further enhance its noise robustness. The proof rested on the condition that the dimension $d$ and factorization rank $r$ in the separable NMF problem coincide with each other. However, it may be unrealistic to expect that the condition holds in separable NMF problems appearing in actual applications; in such problems, $d$ is usually greater than $r$. This paper shows, without the condition $d=r$, that the preconditioned SPA is robust to noise.