Probabilistic symmetries and invariant neural networks

TL;DR

This paper develops a probabilistic framework for understanding and constructing neural networks that are invariant or equivariant under group actions, linking symmetry properties with neural network architecture design.

Contribution

It introduces a probabilistic symmetry perspective, providing a unified theory and generative representations for invariant and equivariant neural networks under compact group actions.

Findings

Provides a complete characterization of neural network structures for invariant distributions

Develops a general program for constructing invariant stochastic and deterministic networks

Shows that recent models are special cases of the proposed framework

Abstract

Treating neural network inputs and outputs as random variables, we characterize the structure of neural networks that can be used to model data that are invariant or equivariant under the action of a compact group. Much recent research has been devoted to encoding invariance under symmetry transformations into neural network architectures, in an effort to improve the performance of deep neural networks in data-scarce, non-i.i.d., or unsupervised settings. By considering group invariance from the perspective of probabilistic symmetry, we establish a link between functional and probabilistic symmetry, and obtain generative functional representations of probability distributions that are invariant or equivariant under the action of a compact group. Our representations completely characterize the structure of neural networks that can be used to model such distributions and yield a general program for constructing invariant stochastic or deterministic neural networks. We demonstrate that examples from the recent literature are special cases, and develop the details of the general program for exchangeable sequences and arrays.