# Control Variate Score Matching for Diffusion Models

arXiv: https://arxiv.org/abs/2512.20003 · HTML: https://mlsanity.com/paper/2512.20003/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 22 of its 44 references that are in the map.
- First posted on arXiv 23 Dec 2025.

## Placed in

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

## Builds on (most cited first, 22 of 22)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Multi-modal Molecule Structure-text Model for Text-based Retrieval and Editing](https://mlsanity.com/paper/2212.10789/) (2022)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [Diffusion Normalizing Flow](https://mlsanity.com/paper/2110.07579/) (2021)
- [Particle Denoising Diffusion Sampler](https://mlsanity.com/paper/2402.06320/) (2024)
- [Target Score Matching](https://mlsanity.com/paper/2402.08667/) (2024)
- [Improved off-policy training of diffusion samplers](https://mlsanity.com/paper/2402.05098/) (2024)
- [Reverse Diffusion Monte Carlo](https://mlsanity.com/paper/2307.02037/) (2023)
- [State-specific protein-ligand complex structure prediction with a multi-scale deep generative model](https://mlsanity.com/paper/2209.15171/) (2022)
- [Stochastic Localization via Iterative Posterior Sampling](https://mlsanity.com/paper/2402.10758/) (2024)
- [Progressive Inference-Time Annealing of Diffusion Models for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2506.16471/) (2025)
- [Guiding Generative Models to Uncover Diverse and Novel Crystals via Reinforcement Learning](https://mlsanity.com/paper/2511.07158/) (2025)
- [Disentangling Total-Variance and Signal-to-Noise-Ratio Improves Diffusion Models](https://mlsanity.com/paper/2502.08598/) (2025)

## Built on by (most cited first, 6 of 6)

- [Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matching](https://mlsanity.com/paper/2603.00530/) (2026)
- [Diffusion Path Samplers via Sequential Monte Carlo](https://mlsanity.com/paper/2601.21951/) (2026)
- [Discrete Adjoint Schr"odinger Bridge Sampler](https://mlsanity.com/paper/2602.08243/) (2026)
- [Variance-Reduced Diffusion Sampling via Target Score Identity](https://mlsanity.com/paper/2601.01594/) (2026)
- [Laplace-Fisher Gate Identities for Optimal Matrix-Gated Blended Score Estimation](https://mlsanity.com/paper/2606.25169/) (2026)
- [Bridging Stochastic Flow Maps and Boltzmann Generators with Normalizing Flows](https://mlsanity.com/paper/2609.31978/) (2026)

## Abstract (arXiv)

Sampling from unnormalized probability densities is a pervasive challenge across the computational and physical sciences. Diffusion models provide a powerful generative framework for this task, but their success relies on accurately estimating the score of the perturbed target distribution. Current approaches face a dichotomy between two standard estimation methods: the Denoising Score Identity (DSI) requires data samples and exhibits high variance at low noise levels, whereas the Target Score Identity (TSI) relies on the energy function and suffers from diverging variance at high noise levels. In this work, we reconcile both approaches by introducing the Control Variate Score Identity (CVSI), an unbiased estimator with an analytically optimal, state- and time-dependent control coefficient that theoretically minimizes variance over the entire diffusion process. CVSI serves as a robust plug-in estimator that significantly enhances performance and efficiency in data-free sampler learning and training-free diffusion sampling. These gains scale to complex, high-dimensional energy-based models.