Deep Sufficient Representation Learning via Mutual Information

TL;DR

This paper introduces MSRL, a mutual information-based method using deep neural networks to learn sufficient data representations, with proven consistency, error bounds, and superior performance in experiments.

Contribution

It presents a novel deep neural network approach for mutual information-based sufficient representation learning with theoretical guarantees and empirical validation.

Findings

MSRL is consistent in approximating conditional densities.

Non-asymptotic error bounds are established for MSRL.

MSRL outperforms existing nonlinear dimension reduction methods.

Abstract

We propose a mutual information-based sufficient representation learning (MSRL) approach, which uses the variational formulation of the mutual information and leverages the approximation power of deep neural networks. MSRL learns a sufficient representation with the maximum mutual information with the response and a user-selected distribution. It can easily handle multi-dimensional continuous or categorical response variables. MSRL is shown to be consistent in the sense that the conditional probability density function of the response variable given the learned representation converges to the conditional probability density function of the response variable given the predictor. Non-asymptotic error bounds for MSRL are also established under suitable conditions. To establish the error bounds, we derive a generalized Dudley's inequality for an order-two U-process indexed by deep neural networks, which may be of independent interest. We discuss how to determine the intrinsic dimension of the underlying data distribution. Moreover, we evaluate the performance of MSRL via extensive numerical experiments and real data analysis and demonstrate that MSRL outperforms some existing nonlinear sufficient dimension reduction methods.