Blind Source Separation via Unsupervised Learning
In this paper, a two-layer neural network is presented that organizes itself to perform blind source separation, i.e. it extracts the unknown independent source signals out of their linear mixtures. The convergence behaviour of the network is analyzed, an
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Abstract In this paper, a two-layer neural network is presented that organizes itself to perform blind source separation, i.e. it extracts the unknown independent source signals out of their linear mixtures. The convergence behaviour of the network is analyzed, and experimental results of separating historical speeches of four different speakers are presented.
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Introduction
Blind source separation (BSS) [2] is aimed at extracting the individual, unobservable, statistically independent source signals out of given linear noisy mixtures of them. BSS is performed as part of independent component analysis (ICA) [1], a novel signal processing technique useful in a variety of applications areas. From the mathematical point of view, BSS is a method to find a separation matrix which transforms the mixture signals into statistically independent signals being good estimations of the original source signals. There are some numerical algorithms [1, 3] using higher order statistics for estimating the separation matrix. Recently, neural networks have been proposed (see [4] for a survey) to perform BSS. The unsupervised neural approaches are based on the observation that introducing a nonlinear function in well known neural networks for principal component analysis (PCA) [5] allows us to perform BSS if the input vector and the distribution of the source signals meet some conditions. This is done in order to deal with the higher order moments required.
G. D. Smith et al., Artificial Neural Nets and Genetic Algorithms © Springer-Verlag Wien 1998
In this paper, an unsupervised BSS learning rule for a two-layer neural network is presented. The learning rule is hierarchical in the sense that output unit i receives contributions from all units j with j < i. The convergence behaviour of the neural network is mathematically analyzed and demonstrated in practice. It will be shown that the network is capable of performing BSS for a set of acoustic source signals, consisting of (excerpts of) historical speeches by Churchill, Kennedy, Luther-King and Armstrong.
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Blind Source Separation
Figure 1 illustrates the steps of an ICA algorithm. The first box represents the unobservable (linear) transformation of the unknown source signal vector s into the observed signal vector iI = M . s. The m x n-mixing matrix M is assumed to have full column rank, i.e. iI has the dimension m, with m 2:
n.
The second box represents a pre whitening step which transforms iI into x = fj-l . pT . iI, such that x has as its covariance matrix Cxx = E(xxT ) the identity matrix 1'n. Prewhitening simplifies the BSS task and can be achieved by any non-neural or neural PCA algorithm [5]. The third (dashed) box represents the BSS part: finding an (orthogonal) separation matrix B to transform x into an output vector if whose components have a maximal degree of statistical independence (measured by so called contrast functions [1]). Each output Yi, i = 1,··· ,n can be regarded as an approximation of one of the source signals ±Sj, j = 1,··· ,n.
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