Theory Seminar, March 29, 2:00pm
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.