# Joint stochastic localization and applications

arXiv: https://arxiv.org/abs/2505.13410 · HTML: https://mlsanity.com/paper/2505.13410/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 54 references that are in the map.
- First posted on arXiv 19 May 2025.

## Builds on (most cited first, 13 of 13)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018)
- [Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial](https://mlsanity.com/paper/2402.07487/) (2024)
- [Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization](https://mlsanity.com/paper/2203.05093/) (2022)
- [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019)
- [Sampling, Diffusions, and Stochastic Localization](https://mlsanity.com/paper/2305.10690/) (2023)
- [Stochastic Localization via Iterative Posterior Sampling](https://mlsanity.com/paper/2402.10758/) (2024)
- [Localization Schemes: A Framework for Proving Mixing Bounds for Markov Chains](https://mlsanity.com/paper/2203.04163/) (2022)
- [Eldan's Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixing](https://mlsanity.com/paper/1612.01507/) (2016)
- [A Slightly Improved Bound for the KLS Constant](https://mlsanity.com/paper/2208.11644/) (2022)
- [Generative diffusion for perceptron problems: statistical physics analysis and efficient algorithms](https://mlsanity.com/paper/2502.16292/) (2025)

## Built on by (most cited first, 2 of 2)

- [Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective](https://mlsanity.com/paper/2601.15500/) (2026)
- [Total Variation Guarantees for Sampling with Stochastic Localization](https://mlsanity.com/paper/2603.29555/) (2026)

## Abstract (arXiv)

Stochastic localization is a pathwise analysis technique that has emerged as a powerful tool in high-dimensional probability and sampling. In this work, we extend stochastic localization to a joint framework for coupling probability measures and explore its applications in distributional data analysis. We first unify existing stochastic localization processes under Eldan's α-scheme and characterize their localization rates. Building on this, we introduce a joint scheme to couple probability measures via concurrent α-schemes driven by a shared Brownian motion. This construction is canonical and induces a family of metrics on the space of probability measures, which we call Eldan's α-distance. Alternative variants that extrapolate optimal Gaussian couplings to log-concave measures are also discussed. We study the theoretical properties of Eldan's α-distance, including its restriction to Gaussian measures and its behavior under affine transformations. For α = 0, we show it is topologically equivalent to the 2-Wasserstein distance for measures supported on a common compact set; we also relate its weighted variants to linearized optimal transport in Wiener space and to score-matching objectives in training diffusion models. Computationally, we develop efficient estimators for Eldan's α-distance in the cases α=0 and α=1/2, with rigorous error guarantees for log-concave and finitely supported measures in the former setting and Gaussian measures in the latter. Finally, we apply Eldan's α-distance as a scalable surrogate for the 2-Wasserstein distance to enable fast pairwise distance estimation and approximate computation of Wasserstein barycenters.