Fast Rank-1 NMF for Missing Data with KL Divergence

TL;DR

This paper introduces A1GM, a fast non-gradient method for rank-1 NMF with missing data, leveraging a new analytical solution for NMMF to improve efficiency over gradient-based approaches.

Contribution

The paper presents a novel analytical formula for rank-1 NMMF and a method (A1GM) that applies this formula to efficiently handle missing data in NMF.

Findings

A1GM outperforms gradient methods in efficiency.

A1GM achieves competitive reconstruction errors.

The analytical formula simplifies solving NMF with missing data.

Abstract

We propose a fast non-gradient-based method of rank-1 non-negative matrix factorization (NMF) for missing data, called A1GM, that minimizes the KL divergence from an input matrix to the reconstructed rank-1 matrix. Our method is based on our new finding of an analytical closed-formula of the best rank-1 non-negative multiple matrix factorization (NMMF), a variety of NMF. NMMF is known to exactly solve NMF for missing data if positions of missing values satisfy a certain condition, and A1GM transforms a given matrix so that the analytical solution to NMMF can be applied. We empirically show that A1GM is more efficient than a gradient method with competitive reconstruction errors.