# Surrogate to Poincar'e inequalities on manifolds for dimension reduction in nonlinear feature spaces

arXiv: https://arxiv.org/abs/2505.01807 · HTML: https://mlsanity.com/paper/2505.01807/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 38 references that are in the map.
- First posted on arXiv 3 May 2025.

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

- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Certified dimension reduction in nonlinear Bayesian inverse problems](https://mlsanity.com/paper/1807.03712/) (2018)
- [Extending classical surrogate modelling to high-dimensions through supervised dimensionality reduction: a data-driven approach](https://mlsanity.com/paper/1812.06309/) (2018)
- [Active Manifolds: A non-linear analogue to Active Subspaces](https://mlsanity.com/paper/1904.13386/) (2019)
- [Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities](https://mlsanity.com/paper/2406.13036/) (2024)

## Built on by (most cited first, 1 of 1)

- [Local EGOP for Continuous Index Learning](https://mlsanity.com/paper/2601.07061/) (2026)

## Abstract (arXiv)

We aim to approximate a continuously differentiable function u:ℝd rightarrow ℝ by a composition of functions fcirc g where g:ℝd rightarrow ℝm, m≤ d, and f : ℝm rightarrow ℝ are built in a two stage procedure. For a fixed g, we build f using classical regression methods, involving evaluations of u. Recent works proposed to build a nonlinear g by minimizing a loss function J(g) derived from Poincar'e inequalities on manifolds, involving evaluations of the gradient of u. A problem is that minimizing J may be a challenging task. Hence in this work, we introduce new convex surrogates to J. Leveraging concentration inequalities, we provide suboptimality results for a class of functions g, including polynomials, and a wide class of input probability measures. We investigate performances on different benchmarks for various training sample sizes. We show that our approach outperforms standard iterative methods for minimizing the training Poincar'e inequality based loss, often resulting in better approximation errors, especially for small training sets and m=1.