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Sufficiently Reduced Distributional Regression

Alexander Henzi, Tiange Liu, Xinwei Shen

arXiv:2609.29291v1stat.MEcs.LGstat.ML

TL;DR

SRDR addresses the challenge of estimating conditional distributions with high-dimensional covariates while recovering a sufficient low-dimensional representation. It jointly learns a reduction and generative predictor through proper-score minimization, and the paper reports consistency in energy distance with asymptotic sufficiency as well as strong empirical performance. The theory has stated scope limits, including no direct coverage of classification settings.

  • Problem

    Conditional-distribution estimation is difficult with multivariate outcomes and high-dimensional covariates, while existing generative regression and nonlinear SDR methods have largely treated sampling and explicit dimension reduction separately.

    Source: §1 · pp. 1–2

  • Method

    SRDR jointly learns a lower-dimensional covariate map and conditional generator by minimizing a strictly proper scoring rule, using the energy score in practice.

    Source: §1 · pp. 2–3

  • Results

    SRDR is consistent in energy distance with asymptotic recovery of sufficient reduction, and empirical applications outperform non-generative SDR methods, with predictive accuracy often superior.

    Source: §5.2 · p. 14, §8 · p. 23

  • Takeaways & Limitations

    The scoring-rule characterization makes sufficient dimension reduction a risk-minimization target that also targets probabilistic predictive accuracy.

    Source: §1 · p. 2

    The consistency result assumes an exact empirical energy-score minimizer, and the general theory does not directly apply to classification settings.

    Source: §8 · p. 23

Abstract

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We propose Sufficiently Reduced Distributional Regression (SRDR), a generative method that combines conditional distribution estimation with nonlinear sufficient dimension reduction (SDR). It builds on a characterization of sufficiency through strictly proper scoring rules: a dimension reduction is sufficient if and only if predicting the response from the reduced covariates incurs no loss in expected score relative to the full covariates. Sufficient dimension reduction thus becomes a risk minimization problem. SRDR jointly trains a dimension reduction map and a generative prediction model by minimizing the energy score, which can be estimated by sampling without density evaluation or adversarial training. The framework extends to multi-environment data and to classification. We prove that the estimated conditional distributions converge in energy distance to the true ones, which implies that the learned representation is asymptotically sufficient. In simulations and applications to CT slice localization, superconductivity, and digit classification, SRDR recovers low-dimensional sufficient structure and matches or outperforms state-of-the-art nonlinear SDR methods in representation quality and predictive performance.

1 Introduction

SRDR combines generative conditional-distribution estimation with nonlinear sufficient dimension reduction, addressing the difficulty of distributional regression with multivariate outcomes and high-dimensional covariates. It characterizes sufficiency through proper scoring rules and estimates a reduced representation and generator jointly, with theoretical and empirical support for the approach.

1 Introduction

  • The combined task remains challenging because high-dimensional covariates complicate conditional-distribution estimation, and multivariate densities or distribution functions can be impractical to calculate and use.Generating samples offers an alternative way to approximate desired quantities empirically.Source: §1 · p. 1
  • Generative conditional-distribution methods and neural nonlinear SDR methods had largely been developed separately, with neither line generally combining sampling with an explicitly recovered low-dimensional representation.The paper notes recent exceptions, including Xu, Yu, and Huang (2025) and concurrent work by Tan, Li, and Xue (2026).Source: §1 · p. 2
  • A reduction is sufficient exactly when its expected strictly proper score matches the expected score from the full covariates, turning SDR into risk minimization.This also targets probabilistic prediction accuracy while reducing dimension.Source: §1 · p. 2
  • SRDR jointly learns a lower-dimensional representation e(X) and generator d(e(X), η) by minimizing an empirical score, with the representation dimension constrained below the covariate dimension.The expected score is minimized when generated conditional outcomes match the true conditional distribution.Source: §1 · p. 3
  • The energy score can be estimated unbiasedly by sampling, without density estimates or adversarial training.With the identity reduction e(x)=x, estimating the generator corresponds to engression.Source: §1 · p. 3
  • Under the stated theory, energy-score minimization consistently estimates conditional distributions in energy distance, implying asymptotic sufficiency; the framework also covers heterogeneous multi-environment data and classification.For multi-environment data, a shared reduction can be sufficient even when covariate marginals and conditional outcome distributions vary across environments.Source: §1 · p. 3
  • Across CT slice localization, superconductivity, and MNIST, SRDR outperforms competing SDR methods in representation quality and predictive accuracy.In transfer learning, a pretrained reduction gives a consistent gain over training the same architecture from scratch, and SRDR is competitive with Ge, Zhou, and Huang (2025).Source: §1 · p. 4

2 Related Literature

Related work spans classical distributional models, flexible machine-learning estimators, and methods for sufficient dimension reduction, but these strands often address different goals. SRDR is situated among methods that combine dimension reduction with conditional sampling, while differing in its scoring-rule formulation and scope.

2 Related Literature

  • Classical distributional approaches combine distribution modeling with dimension reduction but are often limited on complex, high-dimensional data; newer flexible methods typically estimate conditional distributions without explicitly recovering sufficient low-dimensional representations.The passage discusses parametric location-scale-shape models, semiparametric methods, forests, GANs, VAEs, and conditional stochastic interpolation.Source: §2 · p. 4
  • Nonlinear SDR methods use embeddings, forests, or neural networks that optimize dependence measures, but the cited approaches do not generate outcome samples from the reduced representation.The passage also notes that sufficient reductions are non-identifiable up to injective transformations, although a minimal reduction σ-field exists under mild conditions.Source: §2 · p. 5
  • Related methods differ in their modeling choices: Liang, Sun, and Liang (2022) use additive noise and yield a random reduction, while Xu, Yu, and Huang (2025) learn a stochastic interpolation path.The latter approach is described as providing sufficient reduction and consistent conditional-distribution estimation under suitable regularity conditions; SRDR instead targets conditional predictive accuracy and generates samples in one forward pass.Source: §2 · p. 5
  • Engression is the no-reduction special case of SRDR, while distributional principal autoencoders coincide with it when X=Y; the latter setting does not provide a sensible regression-sufficiency target.Tan, Li, and Xue (2026) independently proposed essentially the same method, with a different emphasis on finite-sample rates and parameter efficiency versus SRDR’s general scoring-rule characterization, multi-environment data, and classification.Source: §2 · p. 5

3 Proper Scoring Rules and Sufficiency

The section defines sufficiency through conditional independence and shows that strictly proper scoring rules characterize it by predictive optimality. For the energy score, the gap between reduced- and full-covariate prediction equals a distributional divergence, linking the criterion to conditional-distribution estimation.

3.1 Notation

  • A reduction e(X) is sufficient when Y is conditionally independent of X given e(X), equivalently when their conditional distributions agree almost surely.The identifiable target can be a minimal dimension-reduction σ-field, even though a sufficient representation itself is not unique.Source: §3.1 · p. 6

3.2 Proper Scoring Rules, Divergence, and Entropy

  • A strictly proper scoring rule makes expected score uniquely optimal when the forecast distribution matches the outcome distribution.The energy score is strictly proper for distributions with finite first moment; its associated divergence is nonnegative and vanishes only when the distributions agree.Source: §3.2 · p. 6

3.3 Characterization of Sufficiency via Proper Scoring Rules

  • Expected score for prediction from e(X) equals the full-X minimum if and only if e is sufficient.This model-agnostic result characterizes sufficiency through predictive optimality and applies to any strictly proper scoring rule under the theorem’s assumptions.Source: §3.3 · pp. 6–7

3.4 Sufficient Dimension Reduction with the Energy Score

  • For the energy score, the expected reduced-versus-full score gap equals the expected divergence between conditional distributions and is zero exactly when e is sufficient.The energy-score divergence is energy distance up to a scaling factor of 1/2; unlike Tang and Li (2025+), this formulation estimates the conditional distribution and scales to multivariate outcomes.Source: §3.4 · pp. 7–8

4 Sufficiently Reduced Distributional Regression

SRDR represents the conditional response distribution with a generator driven by a reduced covariate map and independent noise, then fits this representation through scoring-rule minimization. The section describes its optimization, dimension selection, multi-environment extension, and classification formulation.

4.1 Population objective

  • SRDR estimates the conditional distribution with a generator d(e(x), η), whose output distribution is fitted by minimizing an expected scoring-rule loss.The representation dimension q is constrained below the covariate dimension p; the unrestricted existence result allows even q = 1, but practical approximation requires additional function restrictions.Source: §4.1 · p. 9

4.2 Method Description

  • The method approximates the energy score by sampling noise, then uses gradient descent to train ReLU networks for e and d.Gaussian or bounded uniform noise are sensible choices; the bounded option aligns with the consistency assumptions and gives no noticeably different results in practice.Source: §4.2 · pp. 9–10

4.3 Dimension Selection

  • Choose q and noise dimension s using hold-out validation; the out-of-sample score’s elbow identifies a small q after which increasing dimension no longer improves it.The noise dimension is generally less sensitive, and a reasonably large value such as s = 100 is often sufficient.Source: §4.3 · pp. 10–11

4.4 Extension to Multi-Environment Data

  • For heterogeneous environments, SRDR uses one shared reduction e(X) with environment-specific generators d_W, yielding partial sufficiency conditional on W.The scoring-rule objective aggregates error across environments; the resulting estimator’s consistency is established in Appendix C.Source: §4.4 · p. 11

4.5 Extension to Classification Problems

  • For classification, SRDR represents labels as one-hot vectors and trains the same objective to generate probability vectors; its extension is reported competitive with state-of-the-art sufficient dimension reduction methods.A softmax constrains outputs to the probability simplex, and the predicted class is assigned by the largest component of the expected generated vector.Source: §4.5 · p. 12

5 Statistical Guarantees

SRDR’s guarantees connect conditional-distribution estimation to sufficient dimension reduction: under stated assumptions, energy-distance consistency implies asymptotic recovery of a sufficient representation. The results cover multivariate outcomes and multi-environment data, but provide consistency without a rate.

5.1 Assumptions

  • The model assumes outcomes arise from a lower-dimensional map of covariates plus independent noise with a known reference distribution, using iid training data.The assumptions also require bounded support and Lipschitz components for the underlying maps.Source: §5.1 · p. 13
  • The estimator’s consistency analysis assumes energy-score minimization and neural-network classes whose depth or width can grow at a controlled rate.The paper describes assuming a training-data minimizer as strong, while noting this assumption is common in related work.Source: §5.1 · p. 14

5.2 Theorem and Discussion

  • Under the stated assumptions, SRDR is consistent in energy distance for conditional distributions, implying asymptotic recovery of a sufficient dimension reduction.The theorem covers multivariate outcomes and multi-environment data, but gives consistency without a rate.Source: §5.2 · pp. 14–16
  • The energy-distance result does not require a conditional density and extends to multivariate outcomes, but it establishes consistency without a convergence rate.The paper notes compact support also implies convergence in 1-Wasserstein distance.Source: §5.2 · p. 15
  • The true dimensions need not be known exactly: choosing larger representation and noise dimensions still preserves the convergence result.For univariate outcomes, a related rate is available in a frequency-truncated energy distance with a modified training score.Source: §5.2 · p. 15

6 Simulations

Across simulations, SRDR performs well in distributional prediction and recovery of low-dimensional sufficient structure. It improves on engression at smaller sample sizes, while comparisons with nonlinear SDR methods show that reference choice matters when assessing representation alignment.

6.1 Comparison with engression

  • In sparse-linear and sparse-nonlinear simulations, SRDR improves on engression at smaller sample sizes and remains competitive at larger sample sizes when q ≥ q∗.For both variants, test energy score decreases up to q∗ = 3 and then stabilizes; the linear map struggles when the true reduction is nonlinear.Source: §6.1 · p. 17, Figure 1 · p. 17

6.2 Comparison with Competing Methods for Sufficient Dimension Reduction

  • For cos(U), GenSDR and SRDR achieve by far the highest distance correlations and smallest variability across replicates, consistently capturing sufficient structure.The methods have similar distance correlation with U, which does not separate them well in this setting.Source: §6.2 · p. 18, Figure 2 · p. 18
  • Distance correlation with a fixed reference reduction measures geometric alignment, not sufficiency, and its setting-specific maximum may be below 1.That ceiling can hide differences between methods near the attainable upper bound.Source: §6.2 · p. 18
  • Across representation dimensions q, SRDR is the best-performing SDR method in CT localization, remains stable at small dimensions, and provides a full predictive distribution.Figure 3 reports MSE and CRPS for generative methods, and MAE for deterministic methods.Source: Figure 3 · p. 19

7 Applications

Across three held-out applications, SRDR is evaluated against nonlinear SDR methods and full-feature benchmarks, with results spanning predictive accuracy and distributional fit. It performs strongly across the tasks, including at small representation dimensions.

7 Applications

  • Across the fixed train-test split and 10 repeated fits, the applications evaluate predictive performance on held-out data.The CT dataset has 53,500 images from 74 patients, represented by 384-dimensional features, with relative axial slice location as the response.Source: §7 · p. 19, §7.1 · p. 19

7.1 Application to CT Slice Localization

  • SRDR is the best-performing SDR method across q on CT localization, with stable performance at small dimensions and the smallest D2 for both patients at q = 3.The comparison covers MSE and CRPS (or MAE for deterministic methods); SRDR also provides a full predictive distribution.Source: §7.1 · pp. 19–20, Figure 4 · p. 20

7.2 Application to Superconductivity

  • On superconductivity, SRDR is among the best SDR methods for MSE and stabilizes as q increases, while XGBoost has the lowest MSE overall.For distributional evaluation, SRDR and engression have lower CRPS than XGBoost's MAE; at q = 1, SRDR and GenSDR match observed distributional shapes in both thermal-conductivity subgroups.Source: §7.2 · pp. 21–22, Figure 5 · p. 21, Figure 6 · p. 21

7.3 Application to Handwritten Digit Classification (MNIST)

  • On MNIST, SRDR achieves the best or tied-best classification accuracy across q, with especially pronounced improvement at small q.At two dimensions, SRDR and GenSDR produce well-separated clusters for most digits, while some overlap remains for other methods.Source: §7.3 · pp. 22–23, Figure 7 · p. 22, Figure 8 · p. 23

8 Discussion

The discussion presents SRDR as a generative SDR method grounded in strictly proper scoring rules and the energy score. It also identifies limits in the theory and in the evidence for classification and multi-environment settings.

8 Discussion

  • SRDR combines sufficient dimension reduction and distributional regression by minimizing a strictly proper score, targeting predictive accuracy for probabilistic forecasts.The method uses the energy score, adapts engression, and is reported as computationally efficient with statistical consistency guarantees.Source: §8 · p. 23
  • The consistency result assumes an exact empirical energy-score minimizer, while training is nonconvex and validation-based selection of q is not covered by the theory.The general theory covers regression with Lipschitz-continuous reduction and generator, not classification; classification consistency remains for future study.Source: §8 · p. 23
  • The multi-environment extension has consistency guarantees but is illustrated only by a transfer-learning experiment, where a training-environment representation is not necessarily sufficient for a new test environment.Refining the reduction with additional test-environment data shows strong performance, but its theoretical understanding remains limited.Source: §8 · p. 23

A Proof of Lemma 3

This section proves Lemma 3 and develops Theorem 4’s convergence argument through measurable representation and noise transformations, test-function truncation, and error bounds, including a bound on E[S22] that invokes Lemma 10.

A Proof of Lemma 3

  • A Borel isomorphism preserves σ(X), and a measurable transformation of an independent uniform variable constructs noise with distribution ν, yielding a measurable composition.The proof uses Borel isomorphisms between Rp and Rq and between Rs and R to establish these steps.Source: A Proof of Lemma 3 · p. 28

B Proof of Theorem 4

  • Theorem 4’s proof decomposes error into estimation and approximation terms, bounds them separately, and combines the bounds; its main challenges are multivariate outcomes and incorporating noise η.The proof roadmap also includes reformulating energy distance with test functions and restricting its integration domain to a compact set with a small approximation error.Source: B Proof of Theorem 4 · p. 28

B.1 Test functions and Approximation of Energy Distance

  • On compact domains, the sine- and cosine-based test functions are bounded and Lipschitz, and truncating the energy-distance integral approximates it uniformly for distributions with bounded first moments.The truncation error can be made arbitrarily small by choosing the compact domain sufficiently large.Source: §B.1 · pp. 29–30

B.2 Notation

  • The proof targets expected energy distance over the training data, reduces notation to the cosine component because the sine component has equivalent smoothness and boundedness, and treats the fitted maps as empirical-loss minimizers.The authors state that the same proof extends to the overall error L.Source: §B.2 · pp. 30–31

B.3 Decomposition

  • The error decomposition uses an independent copy of the data and bounds the truncation error by restricting integration to R(B), using bounded response support and the compact-domain approximation lemma.The stated additional truncation error is ε/4.Source: §B.3 · p. 31

B.4 Discretizing and bounding E[S21]

  • Under bounded supports for X and η and bounded network weights, the S21 analysis discretizes the integration region and controls the empirical function class through neural-network covering numbers.The derived expectation bound depends on the network-class complexity, grid size, and sample size.Source: §B.4 · pp. 31, 33, 34, 35

B.5 Bounding E[S22]

  • For S22, the proof compares discretized and original criteria using minimizers, Lipschitz continuity of the test function, and a neural-network approximation result for Lipschitz maps.The noise-averaged prediction is expressed through the decoder, and the approximation argument controls the resulting gap.Source: §B.5 · pp. 35–36
  • For the E[S22] bound, the proof sets B_z := max(B_x, M_e + 2L_eB_x√p) and invokes Lemma 10.Source: §B.5 · p. 37

B.6 Completing the Proof

The proof-completion passage combines error-bound terms and gives a sufficient convergence condition. The following appendix section extends the consistency result to multiple environments under analogous assumptions.

B.6 Completing the Proof

  • The error bound includes an approximation term M^-2/m, and the proof states that its displayed sufficient condition makes the error term converge to zero as n grows.The passage also states that two other terms converge to zero when min(N1L1, N2L2) tends to infinity.Source: §B.6 · pp. 37–38

C Consistency in Multi-environment Setting

  • The multi-environment section extends Theorem 4 under assumptions analogous to the single-environment case, with independent samples across environments and equal environment sizes for simplicity.Its model class uses a shared reduction e and environment-specific generators d1 through dK.Source: C Consistency in Multi-environment… · p. 38

C.1 Proof of Theorem 11

The appendix extends the single-environment proof to multiple environments, describes a mixture-loss dimension-selection alternative, and specifies varied synthetic settings.

C.1 Proof of Theorem 11

  • The multi-environment proof applies the approximation argument to a mixture covariate distribution and bounds errors across environments using environment-specific network parameters.The excerpt says the proof then follows the single-environment steps, with rates applying to all K environments.Source: §C.1 · pp. 39–40

D Alternative Dimension Selection Criterion

  • The mixture loss scores allocations of a fixed total input dimension between reduced covariates and noise, enabling selection of q by extracting the first q reduction components.The remaining input dimensions are filled with noise variables.Source: D Alternative Dimension Selection… · p. 40

E.1 Simulation Settings

  • The dense-linear setting uses three projections of 80 covariates, with all covariates contributing to the sufficient representation.Its mean and scale functions reuse those from Section 6.1 with the projections replacing the first three raw coordinates.Source: §E.1 · p. 41
  • Dense nonlinear settings form projections of X and define a nonlinear three-dimensional response representation; the heteroscedastic setting uses a two-dimensional reduction for its response.The heteroscedastic response is generated as Y = f1(X) + f2(X) ε, with independent standard-normal ε.Source: §E.1 · pp. 41–42
  • The simulation settings include multimodal mixtures and correlated nuisance covariates, testing recovery of low-dimensional structure beyond conditional-mean prediction or uncorrelated noise.The mixture response depends on the low-dimensional representation, while nuisance coordinates add no information about Y once the signal coordinates are known.Source: §E.1 · pp. 41–42

E.2 Results

Across four additional synthetic settings, SRDR improves on engression at smaller sample sizes and remains competitive at larger sizes when q is sufficiently large. The results also show a dense-nonlinear misspecification difficulty and competitive performance in heteroscedastic reduction recovery.

E.2 Results

  • SRDR improves on engression at smaller sample sizes and remains competitive at larger sample sizes when q is at least q*, with an elbow pattern except for linear SRDR under dense-nonlinear misspecification.The additional-setting comparison is reported in Figure 9.Source: §E.2 · pp. 42, 44, Figure 9 · p. 43
  • For larger sample sizes in the heteroscedastic setting, SRDR is among the most competitive methods at recovering the two-dimensional sufficient reduction by distance correlation.The comparison evaluates learned reductions against the true reduction on an independent test set.Source: §E.2 · p. 44, Figure 10 · p. 44

E.3 Additional results for Superconductivity

On superconductivity data, the three generative methods have similar prediction-interval performance, while SRDR continues to recover observed distributional features across representation dimensions.

E.3 Additional results for Superconductivity

  • The three methods show similar prediction-interval performance, with no clear advantage for any method; SRDR and GenSDR coverage is slightly below nominal and decreases as q increases.Table 1 reports interval performance across replicates.Source: §E.3 · p. 44
  • Across additional q values, SRDR recovers the main outcome-distribution features, while full-feature engression and XGBoost remain competitive and GMDDNet and BENN often predict more concentrated distributions.Figures 11 and 12 provide the additional histogram comparisons.Source: §E.3 · p. 44

E.4 Additional results for CT Slice Localization

Additional CT results show SRDR generally captures patient-level distribution support and shape, and its prediction intervals are shorter than engression's at similar coverage. In transfer learning, enhanced SRDR has modest positive transfer gaps across four PACS domains, whereas a frozen reduction has negative gaps.

E.4 Additional results for CT Slice Localization

  • SRDR generally captures the support and shape of patient-level CT distributions, while some competing methods produce overly concentrated predictions for smaller q.Figures 13 and 14 show additional choices of q.Source: §E.4 · p. 47, Figure 13 · p. 49, Figure 14 · p. 50
  • SRDR yields shorter CT prediction intervals than engression at similar coverage, while GenSDR tends to overcover with wider intervals.Table 2 reports average coverage and interval widths for 90% and 95% intervals.Source: §E.4 · p. 47, Table 2 · p. 48

E.5 Transfer Learning

  • Enhanced SRDR achieves small positive transfer gaps across all four PACS targets, whereas SRDR-Frozen has substantial negative gaps on every target.The transfer gap compares transfer and scratch accuracy for the same method; scratch accuracies are not directly comparable across methods because their architectures differ.Source: §E.5 · pp. 47–48, Table 3 · p. 51

E.6 Implementation Details

  • For CT prediction, neural reduction methods use a one-hidden-layer width-400 network; SRDR uses a matching-width generative network, and engression serves as a full-feature benchmark.SRDR is trained for 25,000 steps, and engression uses the same energy-score objective on the full feature vector.Source: §E.6 · p. 51

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