By Gustavo Deco, Dragan Obradovic

ISBN-10: 1461240166

ISBN-13: 9781461240167

ISBN-10: 1461284694

ISBN-13: 9781461284697

Neural networks supply a robust new expertise to version and keep an eye on nonlinear and intricate structures. during this booklet, the authors current a close formula of neural networks from the information-theoretic standpoint. They convey how this angle offers new insights into the layout idea of neural networks. specifically they express how those tools will be utilized to the subjects of supervised and unsupervised studying together with function extraction, linear and non-linear autonomous part research, and Boltzmann machines. Readers are assumed to have a easy realizing of neural networks, yet all of the correct innovations from details concept are conscientiously brought and defined. as a result, readers from numerous assorted clinical disciplines, particularly cognitive scientists, engineers, physicists, statisticians, and laptop scientists, will locate this to be a really necessary advent to this topic.

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Input vector. We snaIl assume for simplicity that = O. A translation of the input space to such a coordinate system is always possible. : AN' Proof: We use the method of complete induction on the index k. We begin the proof for the case of k=1. The variance principal component p is given by, a 2l = ai in the direction of normalized first «p.. T··T .. 3) .. T .. 5) Due to the definition of first principal component, the direction p is such that the variance is maximal. s. 7) Applying mathematical induction and assuming that principal components 1 to k - 1 are along the first k - 1 eigenvector directions, the normalized k-th principal component Pk must be perpendicular to the directions of the first k - 1 eigenvectors, meaning that for i = I, ...

7) a = lk = 1 where P is the number of training patterns. Hence, the assumption of the additive Gaussian noise has led to the problem definition which is identical to the standard quadratic error minimization in the completely deterministic setting. Preliminaries of Information Theory and Neural Networks 30 The minimization of E can be performed by different optimization techniques. The most simple one is given by the gradient descent method and this optimization technique defines the learning algorithm called backpropagation.

S. 7) Applying mathematical induction and assuming that principal components 1 to k - 1 are along the first k - 1 eigenvector directions, the normalized k-th principal component Pk must be perpendicular to the directions of the first k - 1 eigenvectors, meaning that for i = I, ... 8) Maximizing the variance cr~ in the normalized k-th principal component direction, the following holds: Proving the result that Pk = if i =k if i ':f. 9) vk and the corresponding variance cr~ o = "A k. e. 2]). 11) where I is the NxN unity matrix and A = [~~o ...

### An Information-Theoretic Approach to Neural Computing by Gustavo Deco, Dragan Obradovic

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