Stochastic Deep Networks

TL;DR

This paper introduces deep neural network architectures capable of processing probability measures and point clouds, addressing challenges like permutation invariance and variable cardinality, with applications in classification, generation, and dynamics prediction.

Contribution

It proposes a novel deep framework for measures that handles permutation invariance, weight variation, and cardinality, bridging measures and Euclidean spaces.

Findings

Effective measure-to-measure mapping demonstrated

Networks show robustness to perturbations

Successful applications in classification and generation tasks

Abstract

Machine learning is increasingly targeting areas where input data cannot be accurately described by a single vector, but can be modeled instead using the more flexible concept of random vectors, namely probability measures or more simply point clouds of varying cardinality. Using deep architectures on measures poses, however, many challenging issues. Indeed, deep architectures are originally designed to handle fixedlength vectors, or, using recursive mechanisms, ordered sequences thereof. In sharp contrast, measures describe a varying number of weighted observations with no particular order. We propose in this work a deep framework designed to handle crucial aspects of measures, namely permutation invariances, variations in weights and cardinality. Architectures derived from this pipeline can (i) map measures to measures - using the concept of push-forward operators; (ii) bridge the gap between measures and Euclidean spaces - through integration steps. This allows to design discriminative networks (to classify or reduce the dimensionality of input measures), generative architectures (to synthesize measures) and recurrent pipelines (to predict measure dynamics). We provide a theoretical analysis of these building blocks, review our architectures' approximation abilities and robustness w.r.t. perturbation, and try them on various discriminative and generative tasks.