Demixing Jazz-Music: Sparse Coding Neural Gas for the Separation of Noisy Overcomplete Sources

Summary

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We consider the problem of separating noisy overcomplete sources from linear mixtures, i.e., we observe N mixtures of M > N sparse sources. We show that the "Sparse Coding Neural Gas" (SCNG) algorithm [8, 9] can be employed in order to estimate the mixing matrix. Based on the learned mixing matrix the sources are obtained by orthogonal matching pursuit. Using synthetically generated data, we evaluate the influence of (i) the coherence of the mixing matrix, (ii) the noise level, and (iii) the sparseness of the sources with respect to the performance that can be achieved on the representation level. Our results show that if the coherence of the mixing matrix and the noise level are sufficiently small and the underlying sources are sufficiently sparse, the sources can be estimated from the observed mixtures. In order to apply our method to real-world data, we try to reconstruct each single instrument of a jazz audio signal given only a two-channel recording. Furthermore, we compare our method to the well-known FastICA [4] algorithm and show that in case of sparse sources and presence of additive noise, our method provides a superior estimation of the mixing matrix.

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Demixing Jazz-Music: Sparse Coding Neural Gas for the Separation of Noisy Overcomplete Sources

(ProQuest: ... denotes formulae omitted.)

1. Introduction

Suppose we are given a number of observations X = (x^sub 1^, ... ,x^sub L^), x^sub j^ ∈ IR^sup N^ that are a linear mixture of a number of sparse sources S = (s^sub 1^,. . . , S^sub m^)^sup T^ = (a^sub 1^, . . . , a^sub L^), S^sub i^ ∈ IR^sup L^ and a^sub j^ ∈ IR^sup M^:

x^sub j^ = Ca^sub j^- + ∈^sub j^ ||∈^sub j^ ≤ δ. (1)

Here C = (c^sub 1^,...,c^sub M^), c^sub j^ ∈ IR^sup N^ denotes the mixing matrix. We require ||c^sub j^|| = 1 without loss of generality. The vector a^sub j^ = (s^sub 1,j^, . . . , S^sub M,j^)^sup T^ contains the contribution of the sources s^sub i^ to the mixture x^sub j^. Additionally, a certain amount of additive noise euroj is present. Is it possible to estimate the sources s^sub i^ only from the mixtures x¿ without knowing the mixing matrix C? In the past, a number of methods have been proposed; these methods can be used to estimate the s¿ and C knowing only the mixtures x^sub j^ assuming ||∈^sub j^|| = 0 and M = N [1]. Some methods assume that the sources are statistically independent [4]. More recently, the problem of estimat...

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