Improved SVD-based Initialization for Nonnegative Matrix Factorization using Low-Rank Correction

TL;DR

This paper introduces NNSVD-LRC, an improved SVD-based initialization method for NMF that reduces initial error, promotes sparsity, and is computationally efficient by leveraging low-rank corrections.

Contribution

The paper proposes NNSVD-LRC, a novel SVD-based NMF initialization that addresses error decrease issues, enhances sparsity, and reduces computational cost compared to existing methods.

Findings

NNSVD-LRC significantly reduces initial error in NMF.

The method provably generates sparse initial factors.

It is faster due to lower-rank SVD computation.

Abstract

Due to the iterative nature of most nonnegative matrix factorization (\textsc{NMF}) algorithms, initialization is a key aspect as it significantly influences both the convergence and the final solution obtained. Many initialization schemes have been proposed for NMF, among which one of the most popular class of methods are based on the singular value decomposition (SVD). However, these SVD-based initializations do not satisfy a rather natural condition, namely that the error should decrease as the rank of factorization increases. In this paper, we propose a novel SVD-based \textsc{NMF} initialization to specifically address this shortcoming by taking into account the SVD factors that were discarded to obtain a nonnegative initialization. This method, referred to as nonnegative SVD with low-rank correction (NNSVD-LRC), allows us to significantly reduce the initial error at a negligible additional computational cost using the low-rank structure of the discarded SVD factors. NNSVD-LRC has two other advantages compared to previous SVD-based initializations: (1) it provably generates sparse initial factors, and (2) it is faster as it only requires to compute a truncated SVD of rank $\lceil r/2 + 1 \rceil$ where $r$ is the factorization rank of the sought NMF decomposition (as opposed to a rank-$r$ truncated SVD for other methods). We show on several standard dense and sparse data sets that our new method competes favorably with state-of-the-art SVD-based initializations for NMF.