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Why do deep neural networks (DNNs) benefit from very high dimensionalparameter spaces? Their huge parameter complexities vs. stunning performancesin practice is all the more intriguing and not explainable using the standardtheory of regular models. In this work, we propose a geometrically flavoredinformation-theoretic approach to study this phenomenon. Namely, we introducethe locally varying dimensionality of the parameter space of neural networkmodels by considering the number of significant dimensions of the Fisherinformation matrix, and model the parameter space as a manifold using theframework of singular semi-Riemannian geometry. We derive model complexitymeasures which yield short description lengths for deep neural network modelsbased on their singularity analysis thus explaining the good performance ofDNNs despite their large number of parameters.

A Geometric Modeling of Occam's Razor in Deep Learning