Despite their great success in practical applications, there is still a lack of theoretical and systematic methods to analyze deep neural networks. In this paper, we illustrate an advanced information theoretic methodology to understand the dynamics of learning and the design of autoencoders, a special type of deep learning architectures that resembles a communication channel. By generalizing the information plane to any cost function, and inspecting the roles and dynamics of different layers using layer-wise information quantities, we emphasize the role that mutual information plays in quantifying learning from data. We further suggest and also experimentally validate, for mean square error training, three fundamental properties regarding the layer-wise flow of information and intrinsic dimensionality of the bottleneck layer, using respectively the data processing inequality and the identification of a bifurcation point in the information plane that is controlled by the given data. Our observations have a direct impact on the optimal design of autoencoders, the design of alternative feedforward training methods, and even in the problem of generalization.
In order to apply this expression to any PDF, Information Theoretic Learning (ITL) @cite_32 uses Parzen-window density estimation with Gaussian kernel @math to estimate @math -norm PDF directly from data. More specifically, the estimator of Renyi's quadratic entropy is given by: where @math represents a realization of @math . Note that this estimator has a free parameter (the kernel size @math ) and that its scalability is constrained by its origin on kernel density estimation @cite_58 .
@cite_58 @cite_32
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Introduction. Survey of Existing Methods. The Kernel Method for Univariate Data. The Kernel Method for Multivariate Data. Three Important Methods. Density Estimation in Action. This book presents the first cohesive treatment of Information Theoretic Learning (ITL) algorithms to adapt linear or nonlinear learning machines both in supervised or unsupervised paradigms. ITL is a framework where the conventional concepts of second order statistics (covariance, L2 distances, correlation functions) are substituted by scalars and functions with information theoretic underpinnings, respectively entropy, mutual information and correntropy. ITL quantifies the stochastic structure of the data beyond second order statistics for improved performance without using full-blown Bayesian approaches that require a much larger computational cost. This is possible because of a non-parametric estimator of Renyis quadratic entropy that is only a function of pairwise differences between samples. The book compares the performance of ITL algorithms with the second order counterparts in many engineering and machine learning applications. Students, practitioners and researchers interested in statistical signal processing, computational intelligence, and machine learning will find in this book the theory to understand the basics, the algorithms to implement applications, and exciting but still unexplored leads that will provide fertile ground for future research.