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An Information-Maximization Approach to Blind Separation and Blind Deconvolution A.J. Bell and T.J. Sejnowski Computational Modeling of Intelligence 11.03.11.(Fri) Summarized by Joon Shik Kim

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Abstract Self-organizing learning algorithm that maximizes the information transferred in a network of nonlinear units. The nonlinearities are able to pick up higher-order moments of the input distribution and perform true redundancy reduction between units in the output representation. We apply the network to the source separaton (or cocktail party) problem, successfully separating unknown mixtures of up to 10 speakers.

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Cocktail Party Problem

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Introduction (1/2) The development of information-theoretic unsupervised learning rules for neural networks The use, in signal processing, of higher- order statistics for separating out mixtures of independent sources (blind separation) or reversing the effect of an unknown filter (blind deconvolution)

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Introduction (2/2) The approach we take to these problem is a generalization of Linsker’s informax principle to nonlinear units with arbitrarily distributed inputs. When inputs are to be passed through a sigmoid function, maximum information transmission can be achieved when the sloping part of the sigmoid is optimally lined up with the high density parts of the inputs. Generation of this rule to multiple units leads to a system that, in maximizing information transfer, also reduces the redundancy between the units in the output layer.

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Information Maximization The basic problem is how to maximize the mutual information that the output Y of a neural network processor contains about its input X. I(Y,X)=H(Y)-H(Y|X)

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Information Maximization Information between inputs and outputs can be maximized by maximizing the entropy of the outputs alone. H(Y|X) tends to minus infinity as the noise variance goes to zero.

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For One Input and One Output When we pass a single input x through a transforming function g(x) to give an output variable y, both I(y,x) and H(y) are maximized when we align high density parts of the probability density function (pdf) of x with highly sloping parts of the function g(x).

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For One Input and One Output

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For an N→N Network

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Inverse of a Matrix

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Blind Separation and Blind Deconvolution

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Blind Separation Results

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Different Aspects from Previous work There is no noise, or rather, there is no noise model in this system. There is no assumption that inputs or outputs have Gaussian statistics. The transfer function is in general nonlinear.

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Conclusion The learning rule is decidedly nonlocal. Each “neuron” must know the cofactor either of all the weights entering it, or all those leaving it. The network rule remains unbiological. We believe that the information maximization approach presented here could serve as a unifying framework that brings together several lines of research, and as a guiding principle for further advances.

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