Sufficient Dimesion Reduction via Generalized Stein's Lemma

Ye Tian

stat.MLAug 15, 2026

Sufficient Dimesion Reduction via Generalized Stein's Lemma

Authors: Ye Tian

Organizations: Key Laboratory of Applied Statistics of MOE, Key Laboratory of Big Data Analysis of Jilin Province, School of Mathematics and Statistics, Northeast Normal University, Renmin Street, Changchun, 130024, Jilin, China

Abstract

Sufficient dimension reduction (SDR) seeks the minimal subspace of the predictors that captures the full conditional distribution of the response, which is known as the central subspace (CS). When the response is multivariate, the problem becomes considerably more challenging, particularly when the sample size is limited. Existing methods face different limitations:inverse regression approaches rely on strong distributional assumptions and matrix inversion, and their multi-response extensions suffer from severe slice sparsity; forward regression methods depend on computationally intensive iterative smoothing whose cost grows with the response dimension; and deep learning-based approaches demand large amounts of labeled data. To circumvent these shortcomings, we propose an SDR framework based on the generalized Stein's lemma. Our method constructs a cross-moment matrix between the multivariate response and the marginal score function of the predictors, and recovers the CS via its singular value decomposition. The proposed method does not rely on the linearity condition, avoids matrix inversion as well as iterative smoothing, and can leverage unlabeled data when available. We establish convergence guarantees for the proposed estimator under standard regularity conditions. Moreover, we propose a practical rank-selection algorithm to estimate the dimension of the CS. Extensive simulation studies and a real data application demonstrate that the proposed methods consistently outperform existing approaches across a variety of settings, particularly in moderate-dimensional, label-scarce scenarios with high noise levels.

Figures & tables

Table 1: Estimation accuracy of the central subspace under different covariate distributions and labeled sample sizes.

Method	Data generating distribution
Method	Isotropic Gaussian			AR(1) Gaussian			Log-normal			Exp-trans AR(1)
	$r=1$	$r=3$	$r=5$	$r=1$	$r=3$	$r=5$	$r=1$	$r=3$	$r=5$	$r=1$	$r=3$	$r=5$
External baselines
SIR	$0.13(0.44)$	$2.91(2.45)$	$4.43(1.08)$	$0.22(0.67)$	$2.96(2.56)$	$4.27(1.34)$	$9.00(0.00)$	$7.00(0.00)$	$5.00(0.00)$	$9.00(0.00)$	$7.00(0.00)$	$5.00(0.00)$
csOPG	$0.28(0.55)$	0.26 (0.44)	$0.59(0.65)$	$0.16(0.42)$	0.29 (0.48)	$0.85(0.61)$	$3.13(1.76)$	$1.88(1.28)$	$1.44(1.17)$	$3.19(1.58)$	$1.94(1.47)$	$1.28(1.21)$
KSIR	$0.09(0.29)$	$0.66(0.65)$	$1.09(0.63)$	$0.07(0.26)$	$0.68(0.68)$	$1.23(0.68)$	$2.26(1.05)$	$3.72(1.88)$	$2.53(1.53)$	$2.20(1.19)$	$4.11(2.06)$	$3.22(1.52)$

Table 2: Rank selection results: absolute error

|\widehat{r}-r|

for each method across four data distributions and three true ranks.

Train ratio	Method	$\gamma$
Train ratio	Method	0.1			0.3			0.5
		$\widehat{r}$	MSE ( $\widehat{r}$ )	MSE ( $r=10$ )	$\widehat{r}$	MSE ( $\widehat{r}$ )	MSE ( $r=10$ )	$\widehat{r}$	MSE ( $\widehat{r}$ )	MSE ( $r=10$ )
0.6	External baselines
	SIR	12.00(0.00)	1.0197(0.1007)	1.0328(0.0893)	11.49(0.50)	0.6795(0.0562)	0.6813(0.0571)	11.00(0.00)	0.6264(0.0515)	0.6257(0.0510)
	csOPG	4.53(1.47)	0.7854(0.0871)	0.7475(0.0732)	5.07(1.42)	0.7571(0.0774)	0.6909(0.0635)	5.59(1.49)	0.7262(0.0726)	0.6785(0.0670)
	KSIR	3.28(1.13)	0.7505(0.0752)	0.7080(0.0708)	3.38(0.88)	0.6992(0.0729)	0.6501(0.0589)	3.36(0.87)	0.6727(0.0666)	0.6186(0.0571)

Table 3: Estimated rank

\widehat{r}

, and test MSE based on

\widehat{r}

and on the reference rank

r=10

, for training proportions 0.6 and 0.8.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Value	Remarks
Optimizer	AdamW	default betas (0.9,0.999)
Learning rate	$1\times 10^{-3}$	–
Weight decay	$1\times 10^{-5}$	–
Batch size	256	–
Epochs	150	fixed, no early stopping
LR scheduler	Cosine annealing	period = 150 epochs

Appendix

Table 4: Training parameters for the denoising score matching network.

Parameter	Value	Remarks
Kernel	curl-free IMQ	curl-free inverse multiquadric
Regularization parameter $\lambda$	1e-4/1e-5	taken to be 1e-4 except $n_{\ell}=500$ in the simulation study
Data standardization	Yes	performed before fitting; gradients rescaled back
Random seed	Set	via np.random.seed() for reproducibility

Appendix

Table 5: Parameters for the TRE.

Parameter	Value	Remarks
Optimizer	Adam	default betas
Learning rate	$1\times 10^{-3}$	–
Weight decay	None	–
Batch size	64	–
Epochs	100	fixed
Loss function	MSE	–

Appendix

Table 6: Training hyperparameters for the MLP baseline.

Parameter	Value	Remarks
Optimizer	Adam	default betas
Learning rate	$1\times 10^{-3}$	–
Batch size	64	–
Epochs	200	early stopping based on validation MSE
Loss function	MSE	–
Gradient clipping	Max norm = 1.0	applied after each backward pass

Appendix

Table 7: Training hyperparameters for the evaluation MLP.

Method	Parameters
$\mathrm{S}_{N,\text{sup}}$ / $\mathrm{S}_{N,\text{semi}}$	Heavy: $\tau=0.95$ ; Light: $\alpha=0.90,m=200$
$\mathrm{S}_{\nu,\text{sup}}$ / $\mathrm{S}_{\nu,\text{semi}}$	Heavy: $\tau=0.95$ ; Light: $\alpha=0.90,m=200$
SIR	Heavy: $\tau=0.95$ ; Light: $\alpha=0.90,m=200$
MLP	Heavy: $\tau_{h}=0.95$ ; Light: $\tau_{l}=0.90,\ \rho=0.85$

Appendix

Table 8: Hyperparameters for rank selection.

Dimensionality Reduction

Dimensionality reduction maps high-dimensional data to compact representations for visualization, prediction, compression, and scientific modeling.

See all 145 papers

Sep 24, 2026

Sufficiently Reduced Distributional Regression

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.

Alexander Henzi, Tiange Liu, Xinwei Shen

Tsinghua University, Department of Statistics and Data Science · University of Washington

Conditional Distribution Estimation Inverse Dimensionality Reduction Dimensionality Reduction Distributional Regression Conditional Generative Modeling

Ask PDF

May 29, 2026

Riemannian Stochastic Optimization for Sufficient Dimension Reduction

Sufficient dimension reduction (SDR) makes high-dimensional regression tractable by projecting the covariates onto a low-dimensional subspace that preserves the conditional mean of the response. Existing gradient-based estimators either operate in the ambient space and suffer from the curse of dimensionality, or localize in the reduced space at a per-outer-iteration cost at least quadratic in the sample size. We show that minimizers of the population Minimum Average Variance Estimation (MAVE) risk approximate the same Grassmannian target as the Outer Product of Gradients (OPG), and recast the empirical criterion as a smooth maximization on the Stiefel manifold with closed-form Riemannian gradient. The resulting algorithm, SMAVE, combines sparse projected-space nearest-neighbor localization with Riemannian stochastic gradient ascent. A simplified version comes with almost-sure convergence and a non-asymptotic rate matching the standard non-convex stochastic first-order scaling. Empirically, SMAVE matches or improves on RMAVE's synthetic subspace recovery at moderate-to-high ambient dimension, and on four real datasets it uniformly improves over OPG and is competitive with or outperforms RMAVE at orders of magnitude lower runtime.

Thibault Pautrel, François Portier

Laboratoire des Signaux et Systèmes (L2S), CentraleSupélec, Université Paris-Saclay, Gif-sur-Yvette, France · ENSAI, CREST, Bruz, France

Stochastic Optimization Riemannian Optimization Inverse Dimensionality Reduction Dimensionality Reduction

Ask PDF

Sep 20, 2026

Belted Engression: Sufficient Dimension Reduction for Generative Distributional Regression

Modern conditional generative models face significant challenges when learning complex covariate dependencies. While sufficient dimension reduction (SDR) provides a principled approach to compress these dependencies, traditional SDR frameworks were not formulated for conditional generation. To bridge this gap, we propose Belted Engression, a unified and architecturally parameter-efficient framework for generative distributional regression. Our approach establishes an end-to-end compress-then-generate paradigm driven by sufficient representation learning, embedding a structural bottleneck into the generative architecture. Theoretically, we prove that the standard SDR condition is equivalent to a law-preserving generative factorization, which is achieved at the global optimum of the population Belted Engression objective. Furthermore, by uncovering a localized Bernstein-type control for the energy-score loss, we establish finite-sample convergence rates that are sharper than those of existing results. We also prove that this belted architecture is strictly smaller, operating with an asymptotically vanishing parameter count relative to the unstructured baseline. Extensive simulations and real-world applications demonstrate that Belted Engression achieves superior distributional prediction and SDR recovery with fewer trainable parameters.

Wenxi Tan, Bing Li, Lingzhou Xue

Department of Statistics, The Pennsylvania State University

Conditional Distribution Estimation Inverse Dimensionality Reduction Dimensionality Reduction Conditional Generative Modeling

Ask PDF

Abstract

Dimensionality Reduction

Explore similar work

Sufficiently Reduced Distributional Regression

Riemannian Stochastic Optimization for Sufficient Dimension Reduction

Belted Engression: Sufficient Dimension Reduction for Generative Distributional Regression