A Quasi-Newton algorithm on the orthogonal manifold for NMF with transform learning
This work addresses a computational bottleneck for researchers and practitioners in audio processing using NMF, though it is incremental as it builds on existing transform learning methods.
The paper tackles the computational overhead of learning transforms in nonnegative matrix factorization (NMF) for audio spectral unmixing by deriving a quasi-Newton method on the orthogonal manifold, which outperforms state-of-the-art methods by orders of magnitude in experiments on synthetic and real audio data.
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.