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.

翻译：我们证明，在$r\leq \ell \leq n/2$参数范围内，随机$2r$一致超图的$\ell$阶Kikuchi图在谱意义上以Sharp（精确至对数因子）的采样率近似完全$2r$一致超图的Kikuchi图。证明基于矩阵Bernstein不等式，但与先前工作不同，我们将其应用于Johnson特征空间的一个恰当分块集合。分析依赖于任意Kikuchi图的一个新的、简单的带状局部性质。作为应用，我们证明当输入为随机超图上具有$\gtrsim n \cdot (n/\ell)^{r-1} \log n$条超边的带噪声植入型$2r$-XOR实例时，Max $2r$-XOR问题的自然$2\ell$次平方和松弛是“整的”。