Nonlinear Blind Source Separation Unifying Vanishing Component Analysis and Temporal Structure

Nonlinear blind source separation (BSS) is one of the unsolved problems in unsupervised learning, because the solutions are highly non-unique when there is no prior information for the mixing functions. In this paper, we present a novel approach to tackle the ill-posedness of the nonlinear BSS problem with a few assumptions. The derivation of our algorithm is inspired by the idea of an efficient layer-by-layer representation to approximate the nonlinearity. Once such a representation is built, a final output layer is constructed by solving a convex optimization problem. Relying on the multi-layer architecture, the algorithm transforms a time-invariant nonlinear BSS to the local linear problem with a tolerable computational cost. Then, the projected data can break the nonlinear problem down into the version of a generalized joint diagonalization problem in the feature space. Importantly, the parameters and forms of polynomials depend solely on the input data, which guarantee the robustness of the structure. We thus address the general problem without being restricted to any specific mixture or parametric model. Experiments show that the proposed algorithm has a higher estimation accuracy on audio data sets from the real world for separating various nonlinear mixtures.