Information-theoretic measures, such as the entropy, the cross-entropy and the Kullback–Leibler divergence between two mixture models, are core primitives in many signal processing tasks. Since the Kullback–Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using costly Monte Carlo stochastic integration, approximated or bounded using various techniques. We present a fast and generic method that builds algorithmically closed-form lower and upper bounds on the entropy, the cross-entropy, the Kullback–Leibler and the α-divergences of mixtures. We illustrate the versatile method by reporting our experiments for approximating the Kullback–Leibler and the α-divergences between univariate exponential mixtures, Gaussian mixtures, Rayleigh mixtures and Gamma mixtures.

We describe a framework to build distances by measuring the tightness of inequalities and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the Hölder ordinary and reverse inequalities and present two novel classes of Hölder divergences and pseudo-divergences that both encapsulate the special case of the Cauchy–Schwarz divergence. We report closed-form formulas for those statistical dissimilarities when considering distributions belonging to the same exponential family provided that the natural parameter space is a cone (e.g., multivariate Gaussians) or affine (e.g., categorical distributions). Those new classes of Hölder distances are invariant to rescaling and thus do not require distributions to be normalized. Finally, we show how to compute statistical Hölder centroids with respect to those divergences and carry out center-based clustering toy experiments on a set of Gaussian distributions which demonstrate empirically that symmetrized Hölder divergences outperform the symmetric Cauchy–Schwarz divergence.

Why do deep neural networks (DNNs) benefit from very high dimensional parameter spaces? Their huge parameter complexities vs . stunning performance in practice is all the more intriguing and not explainable using the standard theory of model selection for regular models. In this work, we propose a geometrically flavored information-theoretic approach to study this phenomenon. With the belief that simplicity is linked to better generalization, as grounded in the theory of minimum description length, the objective of our analysis is to examine and bound the complexity of DNNs. We introduce the locally varying dimensionality of the parameter space of neural network models by considering the number of significant dimensions of the Fisher information matrix, and model the parameter space as a manifold using the framework of singular semi-Riemannian geometry. We derive model complexity measures which yield short description lengths for deep neural network models based on their singularity analysis thus explaining the good performance of DNNs despite their large number of parameters.

Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using costly Monte-Carlo stochastic integration, approximated, or bounded using various techniques. We present a fast and generic method that builds algorithmically closed-form lower and upper bounds on the entropy, the cross-entropy and the Kullback-Leibler divergence of mixtures. We illustrate the versatile method by reporting on our experiments for approximating the Kullback-Leibler divergence between univariate exponential mixtures, Gaussian mixtures, Rayleigh mixtures, and Gamma mixtures.

We propose a new generic type of artificial neurons called q-neurons. A q-neuron is a stochastic neuron with its activation function relying on Jackson's discrete q-derivative for a stochastic parameter q. We show how to generalize neural network architectures with q-neurons and demonstrate the scalability and ease of implementation of q-neurons into legacy deep learning frameworks. We report experimental results that consistently improve performance over state-of-the-art standard activation functions, both on training and test loss functions.