Jacobi Algorithm for Nonnegative Matrix Factorization with Transform Learning

Abstract

Nonnegative matrix factorization (NMF) is the state-of-the-art approach to unsupervised audio source separation. It relies on the factorization of a given short-time frequency transform into a dictionary of spectral patterns and an activation matrix. Recently, we introduced transform learning for NMF (TL-NMF), in which the short-time transform is learnt together with the nonnegative factors. We imposed the transform to be orthogonal likewise the usual Fourier or Cosine transform. TLNMF yields an original non-convex optimization problem over the manifold of orthogonal matrices, for which we proposed a projected gradient descent algorithm in our previous work. In this contribution we describe a new Jacobi approach in which the orthogonal matrix is represented as a randomly chosen product of elementary Givens matrices. The new approach performs favorably as compared to the gradient approach, in particular in terms of robustness with respect to initialization, as illustrated with synthetic and audio decomposition experiments.