Online Nonnegative Matrix Factorization with Outliers

TL;DR

This paper introduces a unified online nonnegative matrix factorization framework capable of handling outliers, with proven convergence and adaptability to various constraints, demonstrating efficiency and effectiveness on multiple large-scale data tasks.

Contribution

The paper presents a novel, systematic framework for online NMF with outlier robustness, including new solvers and convergence proofs, applicable to diverse settings and large-scale data.

Findings

Algorithms are computationally efficient on synthetic and real data.

Effective in tasks like image denoising and shadow removal.

Converges to stationary points with proven almost sure convergence.

Abstract

We propose a unified and systematic framework for performing online nonnegative matrix factorization in the presence of outliers. Our framework is particularly suited to large-scale data. We propose two solvers based on projected gradient descent and the alternating direction method of multipliers. We prove that the sequence of objective values converges almost surely by appealing to the quasi-martingale convergence theorem. We also show the sequence of learned dictionaries converges to the set of stationary points of the expected loss function almost surely. In addition, we extend our basic problem formulation to various settings with different constraints and regularizers. We also adapt the solvers and analyses to each setting. We perform extensive experiments on both synthetic and real datasets. These experiments demonstrate the computational efficiency and efficacy of our algorithms on tasks such as (parts-based) basis learning, image denoising, shadow removal and foreground-background separation.