Multi-dimensional signal separation with Gaussian processes

Gaussian process (GP) models are widely used in machine learning to account for spatial or temporal relationships between multivariate random variables. In this paper, we propose a formulation of underdetermined source separation in multidimensional spaces as a problem involving GP regression. The advantage of the proposed approach is firstly to provide a flexible means to include a variety of prior information concerning the sources and secondly to lead to minimum mean squared error estimates. We show that if the additive GPs are supposed to be locally-stationary, computations can be done very efficiently in the frequency domain. These findings establish a deep connection between GP and nonnegative tensor factorizations with the Itakura-Saito distance and we show that when the signals are monodimensional, the resulting framework coincides with many popular methods that are based on nonnegative matrix factorization and time-frequency masking.