# Sufficient Dimesion Reduction via Generalized Stein's Lemma

arXiv: https://arxiv.org/abs/2608.15121 · HTML: https://mlsanity.com/paper/2608.15121/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 26 references that are in the map.
- First posted on arXiv 15 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [Generalization error bound for denoising score matching under relaxed manifold assumption](https://mlsanity.com/paper/2502.13662/) (2025)
- [Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regression](https://mlsanity.com/paper/2305.00608/) (2023)
- [Tensor Methods for Additive Index Models under Discordance and Heterogeneity](https://mlsanity.com/paper/1807.06693/) (2018)
- [Deep Sufficient Representation Learning via Mutual Information](https://mlsanity.com/paper/2207.10772/) (2022)
- [Multi-Source Transfer Learning of Sparse Single-Index Models](https://mlsanity.com/paper/2606.29658/) (2026)

## Abstract (arXiv)

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.