Majorization-minimization Algorithms for Convolutive NMF with the Beta-divergence

Nonnegative matrix factorization (NMF) has become a method of choice for spectrogram decomposition. However, its inability to capture dependencies across columns of the input motivated the introduction of a variant, convolutive NMF. While algorithms for solving the convolutive NMF problem were previously proposed, they rely on the use of a heuristic that does not insure the convergence of the algorithm (in particular in terms of objective function values). The goal of this work is to propose rigorous update rules, based on a majorization-minimization (MM) approach, for convolutive NMF with the β-divergence (a standard family of measures of fit). Specifically, we derive and study two variants of a convolutive NMF algorithm that are guaranteed to decrease the objective function value at each iteration. The complexity of the algorithms is studied, and the performance in terms of execution time and objective function are evaluated and compared in several numerical experiments using real-world audio data. Experiments show that the proposed MM algorithms consistently provide lower values of the objective function than the heuristic, at similar computational cost.

1. INTRODUCTION

Nonnegative matrix factorization (NMF) consists of decomposing nonnegative data , such as a spectrogram, into $$\begin{equation*}{\mathbf{V}} \approx {\mathbf{WH}}\tag{1}\end{equation*}$$ where and are two nonnegative matrices referred to as dictionary and activation matrix, respectively. K is usually chosen such that the decomposition is low-rank (K < min(M, N)). NMF is known to produce a factorization that gives a part-based representation of V. This method, popularized by Lee and Seung [1], has led to state-of-the-art results in audio source separation [2], [3], [4] and music transcription [5], [6]. In the context of audio processing, NMF is typically applied on a spectrogram, with each column corresponding to one time frame of data. It can lead to a meaningful decomposition where the dictionary tends to capture the vertical structure (spectral patterns) while the activation matrix encodes how these are mixed.