By Gustavo Deco, Dragan Obradovic
Neural networks supply a strong new know-how to version and regulate nonlinear and intricate structures. during this ebook, the authors current a close formula of neural networks from the information-theoretic standpoint. They exhibit how this attitude offers new insights into the layout idea of neural networks. particularly they exhibit how those tools might be utilized to the themes of supervised and unsupervised studying together with characteristic extraction, linear and non-linear autonomous part research, and Boltzmann machines. Readers are assumed to have a easy realizing of neural networks, yet the entire suitable thoughts from info conception are rigorously brought and defined. hence, readers from a number of various medical disciplines, particularly cognitive scientists, engineers, physicists, statisticians, and machine scientists, will locate this to be a really helpful advent to this topic.
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Additional info for An Information-Theoretic Approach to Neural Computing
E. ) with respect to the input ~ with the stochastic signal probability density function is p ( v) . Hence, the goal of the deterministic neural networh is to model the function 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.
Statistical independence implies that the probability distribution is factorizable. Decorrelation (diagonalization of the covariance matrix) and statistical independency are equivalent only in the Gaussian case (see Chapter 4 for more details about this fact). The next section presents an alternative derivation of PC A as the optimal linear compression method. 2 peA and Optimal Reconstruction This section focuses on reconstruction properties of PCA. 13) x where W is a M x N full-row-rank matrix and and y are the Nand M dimensional vectors respectively with N;Z M.
Let be the N-dimensional input vector distributed according to the probability density p (x) . The linear transformation from the N-dimensional input space to an M -dimensional Principal Component space is given by: x Principal Component Analysis: Statistical Approach 43 N Yj = L = 1, ... e. the input defined by: y = wi. Let Qx be the covariance matrix of • • (x-u) • .. 2) u= fp(i)di is the expectation operator and (i) is the mean value of the where ( . ) input vector. We snaIl assume for simplicity that = O.
An Information-Theoretic Approach to Neural Computing by Gustavo Deco, Dragan Obradovic
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