# Belted and Ensembled Neural Network for Linear and Nonlinear Sufficient Dimension Reduction

arXiv: https://arxiv.org/abs/2412.08961 · HTML: https://mlsanity.com/paper/2412.08961/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 45 references that are in the map.
- First posted on arXiv 12 Dec 2024.

## Builds on (most cited first, 8 of 8)

- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Deep Dimension Reduction for Supervised Representation Learning](https://mlsanity.com/paper/2006.05865/) (2020)
- [Nonlinear Sufficient Dimension Reduction with a Stochastic Neural Network](https://mlsanity.com/paper/2210.04349/) (2022)
- [A Kernel-Expanded Stochastic Neural Network](https://mlsanity.com/paper/2201.05319/) (2022)
- [Exploring the Complexity of Deep Neural Networks through Functional Equivalence](https://mlsanity.com/paper/2305.11417/) (2023)
- [Ensemble Conditional Variance Estimator for Sufficient Dimension Reduction](https://mlsanity.com/paper/2102.13435/) (2021)
- [Nonlinear Functional Principal Component Analysis Using Neural Networks](https://mlsanity.com/paper/2306.14388/) (2023)
- [On Sufficient Graphical Models](https://mlsanity.com/paper/2307.04353/) (2023)

## Built on by (most cited first, 4 of 4)

- [On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reduction](https://mlsanity.com/paper/2512.18971/) (2025)
- [Supervised Dynamic Dimension Reduction with Deep Neural Network](https://mlsanity.com/paper/2508.03546/) (2025)
- [Belted Engression: Sufficient Dimension Reduction for Generative Distributional Regression](https://mlsanity.com/paper/2609.23789/) (2026)
- [FlowSDR: Sufficient Dimension Reduction via Conditional Normalizing Flows](https://mlsanity.com/paper/2606.01346/) (2026)

## Abstract (arXiv)

We introduce a unified, flexible, and easy-to-implement framework of sufficient dimension reduction that can accommodate both linear and nonlinear dimension reduction, and both the conditional distribution and the conditional mean as the targets of estimation. This unified framework is achieved by a specially structured neural network -- the Belted and Ensembled Neural Network (BENN) -- that consists of a narrow latent layer, which we call the belt, and a family of transformations of the response, which we call the ensemble. By strategically placing the belt at different layers of the neural network, we can achieve linear or nonlinear sufficient dimension reduction, and by choosing the appropriate transformation families, we can achieve dimension reduction for the conditional distribution or the conditional mean. Moreover, thanks to the advantage of the neural network, the method is very fast to compute, overcoming a computation bottleneck of the traditional sufficient dimension reduction estimators, which involves the inversion of a matrix of dimension either p or n. We develop the algorithm and convergence rate of our method, compare it with existing sufficient dimension reduction methods, and apply it to two data examples.