Nonnegative Matrix Factorization through Cone Collapse

TL;DR

This paper introduces Cone Collapse, a geometric approach to NMF that iteratively shrinks the data cone to its minimal form, leading to improved clustering performance across diverse datasets.

Contribution

It proposes Cone Collapse, a novel geometric algorithm for NMF that explicitly recovers the data cone, and develops CC-NMF, a cone-aware orthogonal NMF model with strong empirical results.

Findings

CC-NMF matches or outperforms existing NMF methods in clustering purity.

The Cone Collapse algorithm finitely recovers the minimal data cone under mild conditions.

Explicit cone recovery enhances NMF clustering effectiveness.

Abstract

Nonnegative matrix factorization (NMF) is a widely used tool for learning parts-based, low-dimensional representations of nonnegative data, with applications in vision, text, and bioinformatics. In clustering applications, orthogonal NMF (ONMF) variants further impose (approximate) orthogonality on the representation matrix so that its rows behave like soft cluster indicators. Existing algorithms, however, are typically derived from optimization viewpoints and do not explicitly exploit the conic geometry induced by NMF: data points lie in a convex cone whose extreme rays encode fundamental directions or "topics". In this work we revisit NMF from this geometric perspective and propose Cone Collapse, an algorithm that starts from the full nonnegative orthant and iteratively shrinks it toward the minimal cone generated by the data. We prove that, under mild assumptions on the data, Cone Collapse terminates in finitely many steps and recovers the minimal generating cone of $\mathbf{X}^\top$ . Building on this basis, we then derive a cone-aware orthogonal NMF model (CC-NMF) by applying uni-orthogonal NMF to the recovered extreme rays. Across 16 benchmark gene-expression, text, and image datasets, CC-NMF consistently matches or outperforms strong NMF baselines-including multiplicative updates, ANLS, projective NMF, ONMF, and sparse NMF-in terms of clustering purity. These results demonstrate that explicitly recovering the data cone can yield both theoretically grounded and empirically strong NMF-based clustering methods.