Concentration Inequalities for Exchangeable Tensors and Matrix-valued Data

2026-01-28

AI Summary

This work addresses the challenge of deriving concentration inequalities for structured tensor and matrix data that are non-independent yet exchangeable—a setting poorly handled by existing methods. Under an exchangeability assumption, the paper establishes Hoeffding- and Bernstein-type tail bounds by introducing a novel framework for exchangeable dependence, which overcomes limitations of Chatterjee’s exchangeable pair approach. The resulting bounds are sharper for combinatorial matrix sums and unify classical results for both independent and exchangeable cases. The analysis integrates tools from exchangeable random variable theory, matrix concentration inequalities, and combinatorial techniques. The theoretical guarantees are validated through applications to average effect estimation in multi-factor response models and fixed-design sketching algorithms in federated learning, with numerical experiments showing excellent agreement with theoretical predictions.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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: Bridging structured and unstructured data

Abstract

We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums. We are interested in tail bounds and concentration inequalities for those structured weighted sums under exchangeability, extending beyond the classical framework of independent terms. We develop Hoeffding and Bernstein bounds provided with structure-dependent exchangeability. Along the way, we recover known results in weighted sum of exchangeable random variables and i.i.d. sums of random matrices to the optimal constants. Notably, we develop a sharper concentration bound for combinatorial sum of matrix arrays than the results previously derived from Chatterjee's method of exchangeable pairs. For applications, the richer structures provide us with novel analytical tools for estimating the average effect of multi-factor response models and studying fixed-design sketching methods in federated averaging. We apply our results to these problems, and find that our theoretical predictions are corroborated by numerical evidence.

Problem

• concentration inequalities
• exchangeable tensors
• matrix-valued data
• tail bounds
• structured weighted sums

Innovation

• concentration inequalities
• exchangeability
• tensor inner products
• matrix-valued data
• combinatorial sums