一个多对数阶的高阶 Cheeger 不等式
A sharp higher-order Cheeger inequality
查看机构详情
- The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文证明了归一化拉普拉斯算子的第 k 个特征值与 k 个不相交集合的最大电导之间存在多对数阶的 Cheeger 不等式，并给出了构造性证明，为谱聚类提供了理论保证。
中文摘要 AI 辅助
设 λ_k(G) 为有限无向加权图的归一化拉普拉斯算子的第 k 个特征值，并设 ρ_G(k) 为 k 个不相交非空顶点集的最小可能最大电导。我们证明 ρ_G(k) ≤ C[1+log(k+1)]^5 √λ_k(G)，其中 C 为绝对常数。该构造恰好给出 k 个集合，并且以 λ_k(G) 为界，所有边界和体积均在原图中度量。更强地，它产生 k 个具有成对不相交支撑的非负函数，其 Rayleigh 商为 O([1+log(k+1)]^{10} λ_k(G))。证明使用了独立的局部截断，其丢失的协方差通过在每个单元中沿主方向的丢失条件来控制。正则化的谱嵌入在整个原始低特征空间上限制截断能量，而自适应构造在每一阶段将剩余系数维度至少减少一半。然后，一个维度论证将几乎秩一的局部协方差转换为恰好 k 个标量见证。
英文摘要
Let $λ_k(G)$ be the $k$th eigenvalue of the normalized Laplacian of a finite undirected weighted graph $G$ with positive degrees, where $k$ is an integer satisfying $1\le k\le |V(G)|$. Let $φ_k(G)$ be the minimum possible maximum conductance of $k$ disjoint nonempty vertex sets. We prove $φ_k(G)\le C\sqrt{λ_k(G)\log(k+1)}$ for an absolute constant $C$. The number of sets and the spectral index are both $k$, and conductance is measured in the original graph. The logarithmic dependence is optimal up to an absolute constant. The proof combines geometric partitioning of the spectral embedding with minimum-cut improvement and adaptive projections in coefficient space. A single conductance threshold is used throughout the construction. The resulting maps have disjoint supports, and the sum of their Gram matrices is bounded below by an absolute positive multiple of the identity. A dyadic maximal estimate bounds the sum of their internal energies uniformly over unit coefficient vectors. A dimension argument using local eigenvalues then yields exactly $k$ disjoint sparse cuts.
