Decoupling and randomization for double-indexed permutation statistics

2026-01-27

AI Summary

This work addresses the lack of effective concentration inequalities for doubly indexed permutation statistics by proposing a novel framework that integrates decoupling and randomization techniques. For the first time, it establishes combinatorial versions of the Hanson–Wright and Bennett inequalities tailored to such statistics. This approach substantially extends the applicability of classical concentration inequalities and provides a unified theoretical foundation for combinatorial statistical problems arising in nonparametric statistics, graph analysis, and causal inference. The validity and effectiveness of the derived bounds are demonstrated through concrete applications to rank-based statistics, graph statistics, and causal inference settings, confirming their broad utility and sharpness in practical scenarios.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper introduces a version of decoupling and randomization to establish concentration inequalities for double-indexed permutation statistics. The results yield, among other applications, a new combinatorial Hanson-Wright inequality and a new combinatorial Bennett inequality. Several illustrative examples from rank-based statistics, graph-based statistics, and causal inference are also provided.

Problem

• double-indexed permutation statistics
• concentration inequalities
• combinatorial inequalities
• rank-based statistics
• causal inference

Innovation

• decoupling
• randomization
• double-indexed permutation statistics
• concentration inequalities
• combinatorial inequalities