- The paper delivers non-asymptotic sample complexity bounds for score matching in polynomial exponential families using curvature concentration and spectral analysis.
- It guarantees recovery of the true exponential family structure and near-optimal parameter estimation with sufficient polynomial samples.
- The analysis shows uniform curvature concentration and polynomial dependence on model dimension, making score-based learning scalable for high-dimensional data.
Finite Sample Analysis for Score Matching in Polynomial Exponential Families
Motivation and Context
Score matching has emerged as a computationally favorable alternative to maximum likelihood estimation for continuous exponential family distributions, especially in settings where the normalizing constant is intractable. While score matching is frequently applied in high-dimensional statistical modeling and structure learning tasks, rigorous non-asymptotic statistical guarantees—particularly finite-sample bounds—have remained elusive. This paper addresses this gap by delivering non-asymptotic sample complexity bounds for structure recovery and parameter estimation within polynomial exponential families, providing crucial insights for both theoretical and practical score-based learning.
Problem Statement and Main Results
The core problem concerns learning the structure and parameters of continuous exponential family distributions with unbounded support, specifically those parameterized via polynomial sufficient statistics. The authors formally analyze the statistical properties of score matching estimators under finite sample regimes, focusing on the scaling of sample complexity with respect to model dimensionality.
The main results include:
- Non-asymptotic sample complexity bounds: The derived bounds exhibit polynomial dependence on the ambient dimension and capture explicit relationships between sample size, model complexity, and the required accuracy for structural recovery.
- Curvature concentration analysis: Rigorous control over the empirical curvature of the score matching loss enables tight finite-sample guarantees by bounding deviations between empirical and population parameters.
- Structure and parameter recovery: The demonstrated bounds guarantee recovery of the true exponential family structure and near-optimality in parameter estimation, given sufficient samples.
These results constitute the first explicit finite-sample guarantees for score matching in such general continuous families, extending beyond previous asymptotic analyses.
Technical Contributions
Curvature Analysis
The paper undertakes a detailed investigation of the score matching loss curvature, leveraging concentration inequalities and spectral analysis to show that empirical curvature matrices converge rapidly to their population counterparts. This theoretical control over curvature is critical for establishing sample complexity results and enables uniform recovery guarantees across the parameter space.
Structure Recovery
The recovery of the exponential family structure—including the identification of nonzero polynomial coefficients—is shown to depend on a polynomial number of samples in model dimension. Sparsity regularization and sharp control of the Hessian structure are exploited to guarantee successful model selection, analogous to classical ℓ1-regularized regression in discrete models but tailored for continuous settings.
Multi-linear Model Recovery
For the special class of multi-linear models (where sufficient statistics are multilinear polynomials), the authors obtain stronger guarantees. In these settings, the finite-sample bounds are notably tighter, allowing for robust recovery under weaker regularity assumptions.
Numerical Results and Claims
The sample complexity bounds are shown to scale polynomially with dimension, an improvement over prior work that either yielded asymptotic statements or exponential scaling dependent on model properties. The paper asserts:
- Polynomial dependence: Recovery is feasible with sample sizes polynomial in the dimension, contradicting claims that score matching may be statistically inefficient in high-dimensional contexts.
- Uniform guarantees: Curvature concentration holds uniformly over the relevant parameter space, allowing for simultaneous control of estimation error and structure selection error.
Implications and Future Directions
Theoretical implications include the establishment of score matching as a statistically efficient method for structure learning in continuous exponential families, opening doors to principled model selection with tractable computational cost. Practically, these results facilitate the deployment of score-based learning in high-dimensional regression, generative modeling (including diffusion models), and structure discovery in scientific data, where continuous distributions of complex structure are ubiquitous.
This analysis invites several avenues for future research: extending finite-sample guarantees to more general classes of models (e.g., mixtures or manifold-constrained families), optimizing regularization protocols for structure recovery, and developing sharper bounds for models with additional symmetry or sparsity. Furthermore, the results provide a foundation for theoretical analysis of the statistical performance of score-based generative models, which are increasingly central in modern machine learning.
Conclusion
This work provides the first rigorous finite-sample analysis for score matching in continuous exponential family distributions parameterized by polynomials, demonstrating polynomial sample complexity and practical structure recovery guarantees. The implications are significant both for high-dimensional statistics and for the theoretical underpinnings of score-based generative modeling, marking a substantial step toward reliable, scalable model selection in continuous domains (2605.14168).