Convergence of Kikuchi matrices to $Γ$-independent and $q$-Gaussian limits

2026-10-01

AI Summary

This study addresses the imprecise spectral norm bounds and the intractability of characterizing the limiting distribution of random Kikuchi matrices in tensor PCA. By leveraging non-commutative probability theory, free probability analysis, and matrix concentration inequalities, this work investigates the asymptotic behavior of high-density random Kikuchi matrices as the dimension grows. It establishes a connection between these matrices and systems of Γ-independent semicircular elements, while further exploring q-Gaussian convergence under a double-limit regime. The primary contribution lies in providing the first complete characterization of the Γ-independent limiting distribution and the q-Gaussian system convergence for Kikuchi matrices. These theoretical advances yield significantly tighter upper bounds on the spectral norm, thereby offering a more rigorous foundation for the algorithmic analysis of tensor methods.

Abstract

Kikuchi matrices are a family of structured matrices that were introduced to study problems involving tensors and hypergraphs. We show that, as the ambient dimension grows, dense random Kikuchi matrices have a limit described by a system of $Γ$-independent semicircular elements. This characterizes their limiting spectral distribution and yields improved bounds on their spectral norm, a key quantity in the analysis of algorithms for Tensor PCA. Finally, we show that, in an appropriate double limit, independent Kikuchi matrices converge to the $q$-Gaussian system, another central object in noncommutative probability.

Problem

• Kikuchi matrices
• limiting spectral distribution
• spectral norm
• Tensor PCA
• q-Gaussian system

Innovation

• Kikuchi matrices
• Gamma-independent semicircular elements
• spectral norm
• Tensor PCA
• q-Gaussian system