# Stein's method for marginals on large graphical models

arXiv: https://arxiv.org/abs/2410.11771 · HTML: https://mlsanity.com/paper/2410.11771/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 84 references that are in the map.
- First posted on arXiv 15 Oct 2024.

## Builds on (most cited first, 24 of 24)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets](https://mlsanity.com/paper/1406.7343/) (2014)
- [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022)
- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [Efficiently learning Ising models on arbitrary graphs](https://mlsanity.com/paper/1411.6156/) (2014)
- [Message Passing Stein Variational Gradient Descent](https://mlsanity.com/paper/1711.04425/) (2017)
- [Learning Graphical Models Using Multiplicative Weights](https://mlsanity.com/paper/1706.06274/) (2017)
- [Density Estimation in Infinite Dimensional Exponential Families](https://mlsanity.com/paper/1312.3516/) (2013)
- [Dimension-independent likelihood-informed MCMC](https://mlsanity.com/paper/1411.3688/) (2014)
- [Stein's Method Meets Computational Statistics: A Review of Some Recent Developments](https://mlsanity.com/paper/2105.03481/) (2021)
- [Likelihood-informed dimension reduction for nonlinear inverse problems](https://mlsanity.com/paper/1403.4680/) (2014)
- [Certified dimension reduction in nonlinear Bayesian inverse problems](https://mlsanity.com/paper/1807.03712/) (2018)
- [Measuring Sample Quality with Diffusions](https://mlsanity.com/paper/1611.06972/) (2016)
- [Estimation of High-Dimensional Graphical Models Using Regularized Score Matching](https://mlsanity.com/paper/1507.00433/) (2015)
- [Beyond normality: Learning sparse probabilistic graphical models in the non-Gaussian setting](https://mlsanity.com/paper/1711.00950/) (2017)
- [MALA-within-Gibbs samplers for high-dimensional distributions with sparse conditional structure](https://mlsanity.com/paper/1908.09429/) (2019)
- [Localization for MCMC: sampling high-dimensional posterior distributions with local structure](https://mlsanity.com/paper/1710.07747/) (2017)
- [Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias](https://mlsanity.com/paper/2408.13115/) (2024)
- [Hierarchical off-diagonal low-rank approximation of Hessians in inverse problems, with application to ice sheet model initializaiton](https://mlsanity.com/paper/2301.03644/) (2023)
- [Uniform error bound for PCA matrix denoising](https://mlsanity.com/paper/2306.12690/) (2023)

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

- [Localized Diffusion Models](https://mlsanity.com/paper/2505.04417/) (2025)
- [A hierarchical entropy method for the delocalization of bias in high-dimensional Langevin Monte Carlo](https://mlsanity.com/paper/2509.08619/) (2025)
- [Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin](https://mlsanity.com/paper/2607.15208/) (2026)

## Abstract (arXiv)

Many spatial models exhibit locality structures that effectively reduce their intrinsic dimensionality, enabling efficient approximation and sampling of high-dimensional distributions. However, existing approximation techniques primarily focus on joint distributions and do not provide precise accuracy control for low-dimensional marginals, which are of primary interest in many practical scenarios. By leveraging the locality structures, we establish a dimension independent uniform error bound for the marginals of approximate distributions. Inspired by the Stein's method, we introduce a novel δ-locality condition that quantifies the locality in distributions, and link it to the structural assumptions such as the sparse graphical models. The theoretical guarantee motivates the localization of existing sampling methods, as we illustrate through the localized likelihood-informed subspace method and localized score matching. We show that by leveraging the locality structure, these methods greatly reduce the sample complexity and computational cost via localized and parallel implementations.