Sufficiently Reduced Distributional Regression

Summary

The paper proposes Sufficiently Reduced Distributional Regression (SRDR), a generative method that unifies nonlinear sufficient dimension reduction (SDR) with conditional distribution estimation. Its core contribution is a characterization of sufficiency via strictly proper scoring rules: e(X) is sufficient if and only if the expected score of predicting Y from P(Y|e(X)) matches that obtained with the full covariates X (Theorem 1, Corollary 2), recasting SDR as standard risk minimization. SRDR jointly trains a reduction map e and a generator d(e(X), eta) by minimizing the energy score, which admits unbiased sampling-based estimates without density evaluation or adversarial training; with the identity reduction this reduces to engression. The framework extends to multi-environment data with a shared reduction and to classification through one-hot responses on the probability simplex. Theorem 4 establishes consistency of the estimated conditionals in energy distance, hence asymptotic sufficiency, under Lipschitzness, bounded support, and an exact empirical-minimizer assumption. Simulations plus CT slice localization, superconductivity, and MNIST show SRDR matching or outperforming GSIR, GMDDNet, StoNet, BENN, and GenSDR, with commendable care over the attainable maximum of the distance-correlation metric (Figure 2); PACS transfer yields only small positive gains over scratch training. Main limitations: no finite-sample rates or optimization-error analysis, heuristic elbow-based dimension selection, single fixed test splits for the real datasets, classification and multi-environment settings outside the theory, and a concurrent, essentially equivalent method (Tan, Li, and Xue 2026).