# A Practical Diffusion Path for Sampling

arXiv: https://arxiv.org/abs/2406.14040 · HTML: https://mlsanity.com/paper/2406.14040/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 26 of its 47 references that are in the map.
- First posted on arXiv 20 Jun 2024.

## Placed in

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

## Builds on (most cited first, 26 of 26)

- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning](https://mlsanity.com/paper/1902.03932/) (2019)
- [On the Importance of Noise Scheduling for Diffusion Models](https://mlsanity.com/paper/2301.10972/) (2023)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [Flow Contrastive Estimation of Energy-Based Models](https://mlsanity.com/paper/1912.00589/) (2019)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Reverse Diffusion Monte Carlo](https://mlsanity.com/paper/2307.02037/) (2023)
- [Blindness of score-based methods to isolated components and mixing proportions](https://mlsanity.com/paper/2008.10087/) (2020)
- [An invitation to sequential Monte Carlo samplers](https://mlsanity.com/paper/2007.11936/) (2020)
- [Stochastic Localization via Iterative Posterior Sampling](https://mlsanity.com/paper/2402.10758/) (2024)
- [Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusion](https://mlsanity.com/paper/2402.17886/) (2024)
- [Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo](https://mlsanity.com/paper/2401.06325/) (2024)
- [Chain of Log-Concave Markov Chains](https://mlsanity.com/paper/2305.19473/) (2023)
- [Kernel Stein Discrepancy thinning: a theoretical perspective of pathologies and a practical fix with regularization](https://mlsanity.com/paper/2301.13528/) (2023)
- [Precision-Recall Divergence Optimization for Generative Modeling with GANs and Normalizing Flows](https://mlsanity.com/paper/2305.18910/) (2023)
- [Convergence Analysis of Schr"odinger-F"ollmer Sampler without Convexity](https://mlsanity.com/paper/2107.04766/) (2021)
- [Fit Like You Sample: Sample-Efficient Generalized Score Matching from Fast Mixing Diffusions](https://mlsanity.com/paper/2306.09332/) (2023)

## Built on by (most cited first, 5 of 5)

- [Learning Boltzmann Generators via Constrained Mass Transport](https://mlsanity.com/paper/2510.18460/) (2025)
- [Score-based deterministic density sampling](https://mlsanity.com/paper/2504.18130/) (2025)
- [Diffusion Path Samplers via Sequential Monte Carlo](https://mlsanity.com/paper/2601.21951/) (2026)
- [Sampling by averaging: A multiscale approach to score estimation](https://mlsanity.com/paper/2508.15069/) (2025)
- [Slithering Through Gaps: Capturing Discrete Isolated Modes via Logistic Bridging](https://mlsanity.com/paper/2604.10821/) (2026)

## Abstract (arXiv)

Diffusion models are state-of-the-art methods in generative modeling when samples from a target probability distribution are available, and can be efficiently sampled, using score matching to estimate score vectors guiding a Langevin process. However, in the setting where samples from the target are not available, e.g. when this target's density is known up to a normalization constant, the score estimation task is challenging. Previous approaches rely on Monte Carlo estimators that are either computationally heavy to implement or sample-inefficient. In this work, we propose a computationally attractive alternative, relying on the so-called dilation path, that yields score vectors that are available in closed-form. This path interpolates between a Dirac and the target distribution using a convolution. We propose a simple implementation of Langevin dynamics guided by the dilation path, using adaptive step-sizes. We illustrate the results of our sampling method on a range of tasks, and shows it performs better than classical alternatives.