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.
As we are going to see this theory allows us to interpret mappings created by arbitrary cost functions which are the norm in deep learning. They are also readily applicable to any dataset because both the ITL estimator of Renyi's entropy and mutual information and their NPD matrix extensions can be directly applied to data. However, there is again a free parameter (the kernel size) that needs to be cross-validated or carefully tuned for the estimation @cite_58 , and the computation complexity is high because of the intrinsic eigendecomposition.
@cite_58
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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.