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.

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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.

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