# Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities

arXiv: https://arxiv.org/abs/2406.13036 · HTML: https://mlsanity.com/paper/2406.13036/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 33 references that are in the map.
- First posted on arXiv 18 Jun 2024.

## Builds on (most cited first, 10 of 10)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priors](https://mlsanity.com/paper/2107.02926/) (2021)
- [When Are Nonconvex Problems Not Scary?](https://mlsanity.com/paper/1510.06096/) (2015)
- [Likelihood-informed dimension reduction for nonlinear inverse problems](https://mlsanity.com/paper/1403.4680/) (2014)
- [Optimal low-rank approximations of Bayesian linear inverse problems](https://mlsanity.com/paper/1407.3463/) (2014)
- [Dimensionally Tight Bounds for Second-Order Hamiltonian Monte Carlo](https://mlsanity.com/paper/1802.08898/) (2018)
- [Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction](https://mlsanity.com/paper/2106.04170/) (2021)
- [Data-Free Likelihood-Informed Dimension Reduction of Bayesian Inverse Problems](https://mlsanity.com/paper/2102.13245/) (2021)
- [Statistical Estimation of the Poincar'e constant and Application to Sampling Multimodal Distributions](https://mlsanity.com/paper/1910.14564/) (2019)
- [Nonlinear Bayesian optimal experimental design using logarithmic Sobolev inequalities](https://mlsanity.com/paper/2402.15053/) (2024)

## Built on by (most cited first, 3 of 3)

- [Expected information gain estimation via density approximations: Sample allocation and dimension reduction](https://mlsanity.com/paper/2411.08390/) (2024)
- [Surrogate to Poincar'e inequalities on manifolds for dimension reduction in nonlinear feature spaces](https://mlsanity.com/paper/2505.01807/) (2025)
- [Gradient-free Riemannian Langevin Sampler](https://mlsanity.com/paper/2607.07519/) (2026)

## Abstract (arXiv)

Identifying low-dimensional structure in high-dimensional probability measures is an essential pre-processing step for efficient sampling. We introduce a method for identifying and approximating a target measure π as a perturbation of a given reference measure μ along a few significant directions of ℝd. The reference measure can be a Gaussian or a nonlinear transformation of a Gaussian, as commonly arising in generative modeling. Our method extends prior work on minimizing majorizations of the Kullback--Leibler divergence to identify optimal approximations within this class of measures. Our main contribution unveils a connection between the dimensional logarithmic Sobolev inequality (LSI) and approximations with this ansatz. Specifically, when the target and reference are both Gaussian, we show that minimizing the dimensional LSI is equivalent to minimizing the KL divergence restricted to this ansatz. For general non-Gaussian measures, the dimensional LSI produces majorants that uniformly improve on previous majorants for gradient-based dimension reduction. We further demonstrate the applicability of this analysis to the squared Hellinger distance, where analogous reasoning shows that the dimensional Poincar'e inequality offers improved bounds.