On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reduction

TL;DR

This paper introduces GenSDR, a generative model-based method for nonlinear sufficient dimension reduction that guarantees full recovery of the underlying structure at both population and sample levels, with broad applicability.

Contribution

The paper proposes GenSDR, a novel generative model approach that ensures exhaustive identification of low-dimensional structures in nonlinear SDR, including extensions for non-Euclidean responses.

Findings

GenSDR fully recovers the central sigma-field at population and sample levels.

The method demonstrates strong empirical performance on complex tasks.

An ensemble extension broadens applicability to non-Euclidean responses.

Abstract

Identifying low-dimensional sufficient structures in nonlinear sufficient dimension reduction (SDR) has long been a fundamental yet challenging problem. Most existing methods lack theoretical guarantees of exhaustiveness in identifying lower dimensional structures, either at the population level or at the sample level. We tackle this issue by proposing a new method, generative sufficient dimension reduction (GenSDR), which leverages modern generative models. We show that GenSDR is able to fully recover the information contained in the central $\sigma$-field at both the population and sample levels. In particular, at the sample level, we establish a consistency property for the GenSDR estimator from the perspective of conditional distributions, capitalizing on the distributional learning capabilities of deep generative models. Moreover, by incorporating an ensemble technique, we extend GenSDR to accommodate scenarios with non-Euclidean responses, thereby substantially broadening its applicability. Extensive numerical results demonstrate the outstanding empirical performance of GenSDR and highlight its strong potential for addressing a wide range of complex, real-world tasks.