Robustness of Minimum-Volume Nonnegative Matrix Factorization under an Expanded Sufficiently Scattered Condition

TL;DR

This paper proves that minimum-volume nonnegative matrix factorization (min-vol NMF) can accurately identify true factors in noisy data if the data is well scattered within a latent simplex, under an expanded sufficient scattering condition.

Contribution

The paper introduces the expanded sufficiently scattered condition and proves robustness of min-vol NMF under this new assumption in noisy environments.

Findings

Min-vol NMF can recover true factors with noise under the new condition.

The expanded sufficiently scattered condition ensures data points are well spread in the latent space.

Theoretical guarantees are provided for the robustness of min-vol NMF.

Abstract

Minimum-volume nonnegative matrix factorization (min-vol NMF) has been used successfully in many applications, such as hyperspectral imaging, chemical kinetics, spectroscopy, topic modeling, and audio source separation. However, its robustness to noise has been a long-standing open problem. In this paper, we prove that min-vol NMF identifies the groundtruth factors in the presence of noise under a condition referred to as the expanded sufficiently scattered condition which requires the data points to be sufficiently well scattered in the latent simplex generated by the basis vectors.