# Sampling with Mollified Interaction Energy Descent

arXiv: https://arxiv.org/abs/2210.13400 · HTML: https://mlsanity.com/paper/2210.13400/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 50 references that are in the map.
- First posted on arXiv 24 Oct 2022.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)
- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [A Non-Asymptotic Analysis for Stein Variational Gradient Descent](https://mlsanity.com/paper/2006.09797/) (2020)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Wasserstein Control of Mirror Langevin Monte Carlo](https://mlsanity.com/paper/2002.04363/) (2020)
- [Efficient constrained sampling via the mirror-Langevin algorithm](https://mlsanity.com/paper/2010.16212/) (2020)
- [Sparse Optimization on Measures with Over-parameterized Gradient Descent](https://mlsanity.com/paper/1907.10300/) (2019)
- [Riemannian Stein Variational Gradient Descent for Bayesian Inference](https://mlsanity.com/paper/1711.11216/) (2017)
- [A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1](https://mlsanity.com/paper/2106.03076/) (2021)
- [Sampling with Mirrored Stein Operators](https://mlsanity.com/paper/2106.12506/) (2021)
- [Dykstra's Algorithm, ADMM, and Coordinate Descent: Connections, Insights, and Extensions](https://mlsanity.com/paper/1705.04768/) (2017)

## Built on by (most cited first, 17 of 17)

- [Theoretical Guarantees for Variational Inference with Fixed-Variance Mixture of Gaussians](https://mlsanity.com/paper/2406.04012/) (2024)
- [Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributions](https://mlsanity.com/paper/2409.20547/) (2024)
- [Constrained Sampling with Primal-Dual Langevin Monte Carlo](https://mlsanity.com/paper/2411.00568/) (2024)
- [GAD-PVI: A General Accelerated Dynamic-Weight Particle-Based Variational Inference Framework](https://mlsanity.com/paper/2312.16429/) (2023)
- [Coin Sampling: Gradient-Based Bayesian Inference without Learning Rates](https://mlsanity.com/paper/2301.11294/) (2023)
- [Kernel Semi-Implicit Variational Inference](https://mlsanity.com/paper/2405.18997/) (2024)
- [Particle Semi-Implicit Variational Inference](https://mlsanity.com/paper/2407.00649/) (2024)
- [Efficient displacement convex optimization with particle gradient descent](https://mlsanity.com/paper/2302.04753/) (2023)
- [A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows](https://mlsanity.com/paper/2507.04330/) (2025)
- [Learning Rate Free Sampling in Constrained Domains](https://mlsanity.com/paper/2305.14943/) (2023)
- [Single-seed generation of Brownian paths and integrals for adaptive and high order SDE solvers](https://mlsanity.com/paper/2405.06464/) (2024)
- [Randomized Midpoint Method for Log-Concave Sampling under Constraints](https://mlsanity.com/paper/2405.15379/) (2024)
- [Monte Carlo with kernel-based Gibbs measures: Guarantees for probabilistic herding](https://mlsanity.com/paper/2402.11736/) (2024)
- [A Kernel Approach for Semi-implicit Variational Inference](https://mlsanity.com/paper/2601.12023/) (2026)
- [Posterior sampling with CNN-based, Plug-and-Play regularization with applications to Post-Stack Seismic Inversion](https://mlsanity.com/paper/2212.14595/) (2022)
- [Functional Gradient Flows for Constrained Sampling](https://mlsanity.com/paper/2410.23170/) (2024)
- [Variational Inference via Entropic Transport Descent](https://mlsanity.com/paper/2606.25265/) (2026)

## Abstract (arXiv)

Sampling from a target measure whose density is only known up to a normalization constant is a fundamental problem in computational statistics and machine learning. In this paper, we present a new optimization-based method for sampling called mollified interaction energy descent (MIED). MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs). These energies rely on mollifier functions -- smooth approximations of the Dirac delta originated from PDE theory. We show that as the mollifier approaches the Dirac delta, the MIE converges to the chi-square divergence with respect to the target measure and the gradient flow of the MIE agrees with that of the chi-square divergence. Optimizing this energy with proper discretization yields a practical first-order particle-based algorithm for sampling in both unconstrained and constrained domains. We show experimentally that for unconstrained sampling problems our algorithm performs on par with existing particle-based algorithms like SVGD, while for constrained sampling problems our method readily incorporates constrained optimization techniques to handle more flexible constraints with strong performance compared to alternatives.