Deep Sufficient Representation Learning via Mutual Information
This work addresses the challenge of sufficient representation learning for researchers in machine learning and statistics, offering a novel method with theoretical guarantees and practical gains, though it is incremental in building on mutual information frameworks.
The paper tackles the problem of learning sufficient representations for multi-dimensional response variables by proposing a mutual information-based approach (MSRL) that ensures consistency and provides non-asymptotic error bounds, demonstrating improved performance over existing nonlinear sufficient dimension reduction methods in experiments.
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.