Stein's method for marginals on large graphical models

2024-10-15

AI Summary

This work addresses the lack of theoretical guarantees for low-dimensional marginal approximation in large graph models. We propose an efficient, structure-aware marginal inference framework grounded in local graph topology. Our key contributions are threefold: (i) We introduce Stein’s method to marginal analysis for the first time, defining a δ-locality condition and deriving dimension-free bounds on marginal approximation error; this condition naturally subsumes common structural assumptions such as sparsity. (ii) We design two localized algorithms—local likelihood-guided subspace projection and local score matching—enabling both sampling and density estimation to be performed locally. (iii) Theoretical analysis and empirical evaluation demonstrate that our approach substantially reduces sample complexity and computational cost, supports parallel implementation, and achieves both high efficiency and accuracy in high-dimensional marginal inference.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Local Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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 mainly focus on joint distributions, and do not guarantee accuracy for low-dimensional marginals. 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 $delta$-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.

Problem

• Ensures accuracy for low-dimensional marginals in high-dimensional distributions
• Introduces δ-locality condition to quantify distribution locality
• Reduces sample complexity via localized, parallel sampling methods

Innovation

• Leverages locality structures for dimension independence
• Introduces δ-locality condition for sparse models
• Localizes sampling methods to reduce complexity