# Finite Sample Bounds for Learning with Score Matching

arXiv: https://arxiv.org/abs/2605.14168 · HTML: https://mlsanity.com/paper/2605.14168/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 29 references that are in the map.
- First posted on arXiv 13 May 2026.

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

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022)
- [Wasserstein Convergence Guarantees for a General Class of Score-Based Generative Models](https://mlsanity.com/paper/2311.11003/) (2023)
- [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)
- [Efficient Learning of Discrete Graphical Models](https://mlsanity.com/paper/1902.00600/) (2019)
- [Generalization error bound for denoising score matching under relaxed manifold assumption](https://mlsanity.com/paper/2502.13662/) (2025)
- [Provable benefits of score matching](https://mlsanity.com/paper/2306.01993/) (2023)
- [Sparse Logistic Regression Learns All Discrete Pairwise Graphical Models](https://mlsanity.com/paper/1810.11905/) (2018)
- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [A Computationally Efficient Method for Learning Exponential Family Distributions](https://mlsanity.com/paper/2110.15397/) (2021)
- [On Learning Continuous Pairwise Markov Random Fields](https://mlsanity.com/paper/2010.15031/) (2020)
- [Computational Implications of Reducing Data to Sufficient Statistics](https://mlsanity.com/paper/1409.3821/) (2014)
- [Learning Continuous Exponential Families Beyond Gaussian](https://mlsanity.com/paper/2102.09198/) (2021)
- [Implicit score matching meets denoising score matching: improved rates of convergence and log-density Hessian estimation](https://mlsanity.com/paper/2512.24378/) (2025)

## Abstract (arXiv)

Learning of continuous exponential family distributions with unbounded support remains an important area of research for both theory and applications in high-dimensional statistics. In recent years, score matching has become a widely used method for learning exponential families with continuous variables due to its computational ease when compared against maximum likelihood estimation. However, theoretical understanding of the statistical properties of score matching is still lacking. In this work, we provide a non-asymptotic sample complexity analysis for learning the structure of exponential families of polynomials with score matching. The derived sample bounds show a polynomial dependence on the model dimension. These bounds are the first of its kind, as all prior work has shown only asymptotic bounds on the sample complexity.