Improved SVD-based initialization for nonnegative matrix factorization using low-rank correction

Abstract

Due to the iterative nature of the most nonnegative matrix factorization (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) and clustering. However, these SVD-based initializations as well as clustering based initializations (if they dense their right factor H), 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 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 other NMF initializations: (1) it provably generates sparse initial factors, and (2) it is faster as it only requires to compute a truncated SVD of rank ⌊r2+1⌋ 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 and clustering based initializations for NMF.