Warren Weaver Hall, Room 312

Higher-Order Cheeger Inequalities

Luca Trevisan, Stanford University

A basic fact of algebraic 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

(non-expanding) 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 cut of

small expansion. We resolve this conjecture. Our result provides a

theoretical justification for clustering algorithms that use the bottom k

eigenvectors to embed the vertices into 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 approximately n/k, and the kth smallest

eigenvalue of the normalized Laplacian matrix, denoted \lambda_k. In

particular, we show that in every graph there are at least k/2 disjoint

sets (one of which will have size at most 2n/k) having each expansion at

most O(\sqrt{\lambda_k \log k}). This result was also discovered

independently by Louis, Raghavendra, Tetali, and Vempala. This bound is

tight, up to constant factors, for the "noisy hypercube" graphs.