A convex formulation for informed source separation in the single channel setting

Blind audio source separation is well-suited for the application of unsupervised techniques such as nonnegative matrix factorization (NMF). It has been shown that on simple examples, it retrieves sensible solutions even in the single-channel setting, which is highly ill-posed. However, it is now widely accepted that NMF alone cannot solve single-channel source separation, for real world audio signals. Several proposals have appeared recently for systems that allow the user to control the output of NMF, by specifying additional equality constraints on the coefficients of the sources in the time-frequency domain. In this article, we show that matrix factorization problems involving these constraints can be formulated as convex problems, using the nuclear norm as a low-rank inducing penalty. We propose to solve the resulting nonsmooth convex formulation using a simple subgradient algorithm. Numerical experiments confirm that the nuclear norm penalty allows the recovery of (approximately) low-rank solutions that satisfy the additional user-imposed constraints. Moreover, for a given computational budget, we show that this algorithm matches the performance or even outperforms state-of-the-art NMF methods in terms of the quality of the estimated sources.