A quasi-Newton algorithm on the orthogonal manifold for NMF with transform learning

Pierre Ablin (1) , Dylan Fagot (2) , Herwig Wendt (3, 4) , Alexandre Gramfort (1, 5) , Cédric Févotte (2, 4)

1 PARIETAL - Modelling brain structure, function and variability based on high-field MRI data
2 IRIT-SC - Signal et Communications
3 IRIT-MINDS - CoMputational imagINg anD viSion
4 CNRS - Centre National de la Recherche Scientifique
5 Inria Saclay - Ile de France

Nonnegative matrix factorization (NMF) is a popular method for audio spectral unmixing. While NMF is traditionally applied to off-the-shelf time-frequency representations based on the short-time Fourier or Cosine transforms, the ability to learn transforms from raw data attracts increasing attention. However, this adds an important computational overhead. When assumed orthogonal (like the Fourier or Cosine transforms), learning the transform yields a non-convex optimization problem on the orthogonal matrix manifold. In this paper, we derive a quasi-Newton method on the manifold using sparse approximations of the Hessian. Experiments on synthetic and real audio data show that the proposed algorithm out-performs state-of-the-art first-order and coordinate-descent methods by orders of magnitude. A Python package for fast TL-NMF is released online at https://github.com/pierreablin/tlnmf.

Mots clés: Non-convex optimization; Nonnegative matrix factorization (NMF); Manifolds; Transform learning; Audio signal processing; Source separation

Domaines: Autres [stat.ML]

Pierre Ablin, Dylan Fagot, Herwig Wendt, Alexandre Gramfort, Cédric Févotte. A quasi-Newton algorithm on the orthogonal manifold for NMF with transform learning. IEEE-ICASSP 2019 - International Conference on Acoustics, Speech and Signal Processing, May 2019, Brighton, United Kingdom. ⟨hal-02346829v2⟩