多路谱划分与高阶Cheeger不等式
Multi-way spectral partitioning and higher-order Cheeger inequalities
摘要：
谱图论中的一个基本事实是：无向图的连通分支数等于该图的拉普拉斯矩阵的特征值零的重数。特别地，图是不连通的当且仅当至少有两个特征值等于零。Cheeger不等式及其变体提供了后一个事实的近似版本;它们指出，当且仅当至少有两个特征值接近于零时，图具有稀疏割。已经证实，类似的表征适用于更高的多重性，即，当且仅当顶点集可以被划分为$k$子集，每个子集定义稀疏割时，存在接近于零的$k$特征值。我们解决这个猜想。我们的结果提供了一个理论上的理由聚类算法，使用底部$k$特征向量嵌入到$\mathbb R^k$的顶点，然后应用几何考虑嵌入。我们还表明，这些技术产生一个接近最佳的折衷之间的扩展集的大小$\approx n/k$，和$k$次最小特征值的归一化拉普拉斯矩阵，表示$λ_k$。特别地，我们证明了在每一个图中，有一个集合的大小最多$2n/k$的扩展最多$O(\sqrt{λ_k \log k})$。这个界是紧的，常数因子，为“噪音超立方体”图。
英文摘要：
A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero. It has been conjectured that an analogous characterization holds for higher multiplicities, i.e., there are $k$ eigenvalues close to zero if and only if the vertex set can be partitioned into $k$ subsets, each defining a sparse cut. We resolve this conjecture. Our result provides a theoretical justification for clustering algorithms that use the bottom $k$ eigenvectors to embed the vertices into $\mathbb R^k$, and then apply geometric considerations to the embedding. We also show that these techniques yield a nearly optimal tradeoff between the expansion of sets of size $\approx n/k$, and the $k$th smallest eigenvalue of the normalized Laplacian matrix, denoted $λ_k$. In particular, we show that in every graph there is a set of size at most $2n/k$ which has expansion at most $O(\sqrt{λ_k \log k})$. This bound is tight, up to constant factors, for the "noisy hypercube" graphs.
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