# Statistical Convergence of Spherical First Hitting Diffusion Models

arXiv: https://arxiv.org/abs/2605.07625 · HTML: https://mlsanity.com/paper/2605.07625/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 15 of its 38 references that are in the map.
- First posted on arXiv 8 May 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 15 of 15)

- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial](https://mlsanity.com/paper/2402.07487/) (2024)
- [Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions](https://mlsanity.com/paper/2402.15602/) (2024)
- [Generalization error bound for denoising score matching under relaxed manifold assumption](https://mlsanity.com/paper/2502.13662/) (2025)
- [First Hitting Diffusion Models for Generating Manifold, Graph and Categorical Data](https://mlsanity.com/paper/2209.01170/) (2022)
- [Nonparametric estimation of a factorizable density using diffusion models](https://mlsanity.com/paper/2501.01783/) (2025)
- [Statistical guarantees for denoising reflected diffusion models](https://mlsanity.com/paper/2411.01563/) (2024)
- [Beyond Fixed Horizons: A Theoretical Framework for Adaptive Denoising Diffusions](https://mlsanity.com/paper/2501.19373/) (2025)
- [Reflected diffusion models adapt to low-dimensional data](https://mlsanity.com/paper/2603.24495/) (2026)

## Abstract (arXiv)

Denoising diffusion models have evolved into a state-of-the-art method for tasks in various fields, such as denoising and generation of images, text generation, or generation of synthetic data for training of other machine learning models. First hitting diffusion models (FHDM) are a particular class of denoising diffusion models with textitrandom adaptive generation time tailored to generate data on a known manifold. Building on the conditioning framework of Doob's h-transform these models leverage the given information on the target data manifold to demonstrate strong performance across tasks while offering distinct features such as time-homogeneous dynamics of the generating process and a reduced average simulation time. Even though the theoretical investigation of standard forward-backward diffusion models has attracted much attention in the recent past, the statistical convergence properties of FHDMs are not yet understood. In this work, we show that, up to logarithmic factors, FHDMs achieve the minimax optimal convergence rate in total variation for spherically supported Sobolev smooth data distributions. In particular, this is the first statistical optimality result for denoising diffusion modelling with random generation time.