# FlowSDR: Sufficient Dimension Reduction via Conditional Normalizing Flows

arXiv: https://arxiv.org/abs/2606.01346 · HTML: https://mlsanity.com/paper/2606.01346/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 51 references that are in the map.
- First posted on arXiv 31 May 2026.

## Builds on (most cited first, 20 of 20)

- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Faithful Heteroscedastic Regression with Neural Networks](https://mlsanity.com/paper/2212.09184/) (2022)
- [Triangular Flows for Generative Modeling: Statistical Consistency, Smoothness Classes, and Fast Rates](https://mlsanity.com/paper/2112.15595/) (2021)
- [Sufficient dimension reduction for classification using principal optimal transport direction](https://mlsanity.com/paper/2010.09921/) (2020)
- [Nonlinear Sufficient Dimension Reduction with a Stochastic Neural Network](https://mlsanity.com/paper/2210.04349/) (2022)
- [Matching Using Sufficient Dimension Reduction for Causal Inference](https://mlsanity.com/paper/1702.00444/) (2017)
- [Belted and Ensembled Neural Network for Linear and Nonlinear Sufficient Dimension Reduction](https://mlsanity.com/paper/2412.08961/) (2024)
- [High-dimensional Interactions Detection with Sparse Principal Hessian Matrix](https://mlsanity.com/paper/1901.07970/) (2019)
- [On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reduction](https://mlsanity.com/paper/2512.18971/) (2025)
- [Conditional Density Estimation with Dimensionality Reduction via Squared-Loss Conditional Entropy Minimization](https://mlsanity.com/paper/1404.6876/) (2014)
- [Golden Ratio-Based Sufficient Dimension Reduction](https://mlsanity.com/paper/2410.19300/) (2024)

## Abstract (arXiv)

Sufficient dimension reduction (SDR) seeks a low-dimensional linear projection of predictors that preserves the conditional distribution of the response. Existing methods target this conditional distribution indirectly, via inverse moments, local forward regression, or neural ensemble regression. We propose FlowSDR, a likelihood-based framework that jointly learns the projection and the conditional density by maximizing a conditional log-likelihood, with the density parameterized by monotone rational-quadratic spline flows. The estimator is Fisher consistent under the SDR model, and its sample objective admits a population interpretation in terms of mutual information. As a complementary model within the same likelihood framework, we introduce the neural Gaussian SDR, a heteroscedastic conditional Gaussian model whose mean and variance are parameterized by shared neural-network functions of the projected predictors. In simulations spanning Gaussian errors, heavy-tailed distributions, two-component mixtures, and settings with tail behavior not captured by mean-variance structure, FlowSDR recovers the central subspace more accurately than existing SDR methods and the neural Gaussian SDR baseline. We further validate these advantages on a face-age prediction task using the UTKFace dataset.