Rank-One NMF-Based Initialization for NMF and Relative Error Bounds under a Geometric Assumption

TL;DR

This paper introduces a geometric assumption-based initialization method for NMF that provides error bounds, improves speed, and enhances clustering performance on various datasets.

Contribution

It presents a novel geometric assumption enabling exact clustering and fast NMF initialization with theoretical error bounds and practical advantages.

Findings

Comparable relative errors to classical NMF algorithms

Faster computation times for the proposed method

Improved clustering performance when combined with classical NMF algorithms

Abstract

We propose a geometric assumption on nonnegative data matrices such that under this assumption, we are able to provide upper bounds (both deterministic and probabilistic) on the relative error of nonnegative matrix factorization (NMF). The algorithm we propose first uses the geometric assumption to obtain an exact clustering of the columns of the data matrix; subsequently, it employs several rank-one NMFs to obtain the final decomposition. When applied to data matrices generated from our statistical model, we observe that our proposed algorithm produces factor matrices with comparable relative errors vis-\`a-vis classical NMF algorithms but with much faster speeds. On face image and hyperspectral imaging datasets, we demonstrate that our algorithm provides an excellent initialization for applying other NMF algorithms at a low computational cost. Finally, we show on face and text datasets that the combinations of our algorithm and several classical NMF algorithms outperform other algorithms in terms of clustering performance.