Differentiable Dictionary Search: Integrating Linear Mixing with Deep Non-Linear Modelling for Audio Source Separation

This paper describes several improvements to a new method for signal decomposition that we recently formulated under the name of Differentiable Dictionary Search (DDS). The fundamental idea of DDS is to exploit a class of powerful deep invertible density estimators called normalizing flows, to model the dictionary in a linear decomposition method such as NMF, effectively creating a bijection between the space of dictionary elements and the associated probability space, allowing a differentiable search through the dictionary space, guided by the estimated densities. As the initial formulation was a proof of concept with some practical limitations, we will present several steps towards making it scalable, hoping to improve both the computational complexity of the method and its signal decomposition capabilities. As a testbed for experimental evaluation, we choose the task of frame-level piano transcription, where the signal is to be decomposed into sources whose activity is attributed to individual piano notes. To highlight the impact of improved non-linear modelling of sources, we compare variants of our method to a linear overcomplete NMF baseline. Experimental results will show that even in the absence of additional constraints, our models produce increasingly sparse and precise decompositions, according to two pertinent evaluation measures.

Probabilistic Modelling of Signal Mixtures with Differentiable Dictionaries

We introduce a novel way to incorporate prior information into (semi-) supervised non-negative matrix factorization, which we call differentiable dictionary search. It enables general, highly flexible and principled modelling of mixtures where non-linear sources are linearly mixed. We study its behavior on an audio decomposition task, and conduct an extensive, highly controlled study of its modelling capabilities.