Two Algorithms for Orthogonal Nonnegative Matrix Factorization with Application to Clustering

TL;DR

This paper introduces two novel algorithms for orthogonal nonnegative matrix factorization (ONMF), improving clustering applications by providing more effective solutions with theoretical insights and practical comparisons.

Contribution

The paper presents two new methods for solving ONMF, including an EM-like algorithm based on spherical k-means equivalence and an augmented Lagrangian approach with strict orthogonality enforcement.

Findings

Both methods outperform standard ONMF algorithms on synthetic, text, and image datasets.

The EM-like algorithm clarifies when to prefer ONMF over k-means.

The augmented Lagrangian approach effectively enforces orthogonality at each step.

Abstract

Approximate matrix factorization techniques with both nonnegativity and orthogonality constraints, referred to as orthogonal nonnegative matrix factorization (ONMF), have been recently introduced and shown to work remarkably well for clustering tasks such as document classification. In this paper, we introduce two new methods to solve ONMF. First, we show athematical equivalence between ONMF and a weighted variant of spherical k-means, from which we derive our first method, a simple EM-like algorithm. This also allows us to determine when ONMF should be preferred to k-means and spherical k-means. Our second method is based on an augmented Lagrangian approach. Standard ONMF algorithms typically enforce nonnegativity for their iterates while trying to achieve orthogonality at the limit (e.g., using a proper penalization term or a suitably chosen search direction). Our method works the opposite way: orthogonality is strictly imposed at each step while nonnegativity is asymptotically obtained, using a quadratic penalty. Finally, we show that the two proposed approaches compare favorably with standard ONMF algorithms on synthetic, text and image data sets.