# Sampling from Energy distributions with Target Concrete Score Identity

arXiv: https://arxiv.org/abs/2510.23106 · HTML: https://mlsanity.com/paper/2510.23106/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 27 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Builds on (most cited first, 19 of 19)

- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022)
- [Rotary Position Embedding for Vision Transformer](https://mlsanity.com/paper/2403.13298/) (2024)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Simple Guidance Mechanisms for Discrete Diffusion Models](https://mlsanity.com/paper/2412.10193/) (2024)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)
- [Concrete Score Matching: Generalized Score Matching for Discrete Data](https://mlsanity.com/paper/2211.00802/) (2022)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Oops I Took A Gradient: Scalable Sampling for Discrete Distributions](https://mlsanity.com/paper/2102.04509/) (2021)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [Target Score Matching](https://mlsanity.com/paper/2402.08667/) (2024)
- [LEAPS: A discrete neural sampler via locally equivariant networks](https://mlsanity.com/paper/2502.10843/) (2025)
- [Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics](https://mlsanity.com/paper/2502.08696/) (2025)
- [Target Concrete Score Matching: A Holistic Framework for Discrete Diffusion](https://mlsanity.com/paper/2504.16431/) (2025)
- [Discrete Neural Flow Samplers with Locally Equivariant Transformer](https://mlsanity.com/paper/2505.17741/) (2025)
- [Reheated Gradient-based Discrete Sampling for Combinatorial Optimization](https://mlsanity.com/paper/2503.04047/) (2025)

## Built on by (most cited first, 3 of 3)

- [Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matching](https://mlsanity.com/paper/2603.00530/) (2026)
- [Discrete Adjoint Schr"odinger Bridge Sampler](https://mlsanity.com/paper/2602.08243/) (2026)
- [MetaDNS: Enhancing Exploration in Discrete Neural Samplers via Well-Tempered Metadynamics](https://mlsanity.com/paper/2605.21722/) (2026)

## Abstract (arXiv)

We introduce the Target Concrete Score Identity Sampler (TCSIS), a method for sampling from unnormalized densities on discrete state spaces by learning the reverse dynamics of a Continuous-Time Markov Chain (CTMC). Our approach builds on a forward in time CTMC with a uniform noising kernel and relies on the proposed Target Concrete Score Identity, which relates the concrete score, the ratio of marginal probabilities of two states, to a ratio of expectations of Boltzmann factors under the forward uniform diffusion kernel. This formulation enables Monte Carlo estimation of the concrete score without requiring samples from the target distribution or computation of the partition function. We approximate the concrete score with a neural network and propose two algorithms: Self-Normalized TCSIS and Unbiased TCSIS. Finally, we demonstrate the effectiveness of TCSIS on problems from statistical physics.