Use of PALM for ℓ₁ sparse matrix factorization: Difficulty and rationalization of a two-step approach

Blind Source Separation (BSS) is a key machine learning method, which has been successfully applied to analyze multichannel data in various domains ranging from medical imaging to astrophysics. Being an ill-posed matrix factorization problem, it is necessary to introduce extra regularizing priors on the sources. While using sparsity has led to improved factorization results, the quality of the separation process turns out to be dramatically dependent on the minimization strategy and the regularization parameters. In this scope, the Proximal Alternating Linearized Minimization (PALM) has recently attracted a lot of interest as a generic, fast and highly flexible algorithm. Using PALM for sparse BSS is theoretically well grounded, but getting good empirical results requires a fine tuning of the involved regularization parameters, which might be too computationally expensive with real-world large-scale data, therefore mandating automatic parameter choice strategies. In this article, we first investigate the empirical limitations of using the PALM algorithm to perform sparse BSS and we explain their origin. Based on this, we further study and justify an alternative two-step algorithmic framework combining PALM with a heuristic approach, namely the Generalized Morphological Component Analysis (GMCA). This method enables an automatic parameter choice for the PALM step. Numerical experiments with comparisons to standard algorithms are carried out on two realistic experiments in spectroscopy and astrophysics.