Robustness Analysis of Hottopixx, a Linear Programming Model for Factoring Nonnegative Matrices

TL;DR

This paper analyzes the robustness of the Hottopixx linear programming model for nonnegative matrix factorization, introducing a more resilient variant that handles duplicates and near duplicates effectively.

Contribution

The paper provides a new, more general robustness analysis of Hottopixx and proposes an improved variant with post-processing to manage duplicates.

Findings

Enhanced robustness of Hottopixx against data duplicates

The proposed method effectively handles near duplicates in datasets

The new variant demonstrates improved stability in factorization results

Abstract

Although nonnegative matrix factorization (NMF) is NP-hard in general, it has been shown very recently that it is tractable under the assumption that the input nonnegative data matrix is close to being separable (separability requires that all columns of the input matrix belongs to the cone spanned by a small subset of these columns). Since then, several algorithms have been designed to handle this subclass of NMF problems. In particular, Bittorf, Recht, R\'e and Tropp (`Factoring nonnegative matrices with linear programs', NIPS 2012) proposed a linear programming model, referred to as Hottopixx. In this paper, we provide a new and more general robustness analysis of their method. In particular, we design a provably more robust variant using a post-processing strategy which allows us to deal with duplicates and near duplicates in the dataset.