Sparse and Unique Nonnegative Matrix Factorization Through Data Preprocessing

TL;DR

This paper introduces a novel data preprocessing method for nonnegative matrix factorization (NMF) that enhances solution uniqueness and sparsity, leveraging M-matrix theory and geometric insights, especially under the separability assumption.

Contribution

It proposes a new preprocessing approach to make NMF more well-posed and sparse, with theoretical guarantees and finite factorizations for rank-three matrices.

Findings

Proven to produce optimal sparse solutions under separability

Makes the number of exact factorizations finite for rank-three matrices

Effective on several image datasets

Abstract

Nonnegative matrix factorization (NMF) has become a very popular technique in machine learning because it automatically extracts meaningful features through a sparse and part-based representation. However, NMF has the drawback of being highly ill-posed, that is, there typically exist many different but equivalent factorizations. In this paper, we introduce a completely new way to obtaining more well-posed NMF problems whose solutions are sparser. Our technique is based on the preprocessing of the nonnegative input data matrix, and relies on the theory of M-matrices and the geometric interpretation of NMF. This approach provably leads to optimal and sparse solutions under the separability assumption of Donoho and Stodden (NIPS, 2003), and, for rank-three matrices, makes the number of exact factorizations finite. We illustrate the effectiveness of our technique on several image datasets.