Kikuchi层级对$k$XOR是紧的
The Kikuchi Hierarchy is Sharp for $k$XOR
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中文总结 AI 辅助
针对稀疏$k$XOR的谱分析存在多对数因子损失的问题，提出归一化Kikuchi层级实现紧的信号-时间权衡，给出匹配下界与量子加速，并用于证明超图Moore界猜想。
中文摘要 AI 辅助
植入噪声$k$XOR与随机$k$XOR的强反驳受一个关于信号强度和时间的猜想权衡支配：Kikuchi层级的第$\text{ell}$层应达到平滑曲线\begin{equation*}m\gtrsim\rho^{-2}n^{k/2}/\beta^{k/2-1}\text{个子句}\text{等价于}\text{可在时间}n^{O(\text{ell})}\text{内求解}\text{其中}\rho\text{是植入信号的偏置，对反驳而言则是目标优势。}\text{然而，目前所有对稀疏}k\text{XOR的谱分析都比这条曲线损失多对数因子，这一损失会进入运行时间的指数项。}\text{我们证明，Kikuchi层级的一个归一化变体在所有元数}k\text{≥}\text{3时都能达到紧的猜想权衡，没有对数损失。在上述尺度下，我们的算法实现了强检测、弱恢复和强反驳；额外的清理步骤可将弱恢复提升为精确恢复，且反驳证书生成度数为}O_k(\text{ell})\text{的平方和证明。我们还在同一模型中证明了匹配的下界。推理和反驳的上界可推广到更一般的植入分布和谓词。最后，我们提出了一种量子算法，在检测和弱恢复任务上比经典谱算法实现四次方加速。}\text{证明基于两个关键要素：稀疏Kikuchi矩阵的归一化，以及其迹展开中闭游走的紧计数。我们在一篇配套论文中使用密切相关的迹-游走计数证明了Feige 2008年的超图Moore界猜想。
英文摘要
Planted noisy $k$XOR and the strong refutation of random $k$XOR are governed by a conjectured trade-off between signal strength and time: Level $\ell$ of the Kikuchi hierarchy should achieve the smooth curve \begin{equation*} m\ \gtrsim\ ρ^{-2}n^{k/2}/\ell^{k/2-1}\ \text{clauses} \quad\Longleftrightarrow\quad \text{solvable in time }n^{O(\ell)}, \end{equation*} where $ρ$ is the bias of the planted signal or, for refutation, the target advantage. However, every spectral analysis of sparse $k$XOR to date loses polylogarithmic factors against this curve, a loss that enters the exponent of the running time. We show that a normalized variant of the Kikuchi hierarchy achieves the sharp conjectured trade-off, with no logarithmic loss, at every arity $k\ge3$. At the scale above, our algorithms achieve strong detection, weak recovery, and strong refutation; an additional cleanup step boosts weak recovery to exact recovery, and the refutation certificates yield sum-of-squares proofs of degree $O_k(\ell)$. We also prove matching lower bounds in the same model. The inference and refutation upper bounds transfer to more general planting laws and predicates. Finally, we give a quantum algorithm that achieves a quartic speedup over the classical spectral algorithms for detection and weak recovery. The proofs rest on two key ingredients: a normalization of the sparse Kikuchi matrix, and a sharp count of the closed walks in its trace expansion. We use a closely related trace-walk count to prove Feige's 2008 hypergraph Moore bound conjecture in a companion paper.
