Kikuchi Graphs of Random Hypergraphs are Approximately Johnson

2026-06-07

AI Summary

This work investigates the spectral approximation of the ℓ-th order Kikuchi graph of a random 2r-uniform hypergraph to that of the complete hypergraph, with applications to the Max 2r-XOR problem. By introducing a novel analysis framework incorporating locality, the authors apply the matrix Bernstein inequality for the first time to the block structure of the Johnson scheme, establishing tight spectral approximation bounds. They prove that when the number of hyperedges is at least \( n \cdot (n/\ell)^{r-1} \cdot \log n \), the degree-2ℓ Sum-of-Squares (SoS) semidefinite relaxation is integral on noisy planted 2r-XOR instances, achieving an approximation bound that is optimal up to logarithmic factors.

Abstract

We prove that level-$\ell$ Kikuchi graphs of random $2r$-uniform hypergraphs spectrally approximate the Kikuchi graph of the complete $2r$-uniform hypergraph at a sampling rate that is sharp up to a logarithmic factor, in the regime $r\leq \ell \leq n/2$. Our proof is based on the matrix Bernstein inequality, but, unlike prior works, we apply it to an appropriate collection of blocks of Johnson eigenspaces. Our analysis relies on a new, simple band-locality property for arbitrary Kikuchi graphs. As an application, we prove that the natural degree-$2\ell$ sum-of-squares relaxation for the Max $2r$-XOR problem is ``integral'' when the input is a planted noisy $2r$-XOR instance on a random hypergraph with $\gtrsim n \cdot (n/\ell)^{r-1} \log n$ hyperedges.

Problem

• Kikuchi graphs
• random hypergraphs
• spectral approximation
• Johnson graphs
• Max 2r-XOR

Innovation

• Kikuchi graphs
• Johnson scheme
• matrix Bernstein inequality
• band-locality
• sum-of-squares relaxation