59 research outputs found
Multi-way spectral partitioning and higher-order Cheeger inequalities
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 eigenvalues close to zero if and only if
the vertex set can be partitioned into subsets, each defining a sparse cut.
We resolve this conjecture. Our result provides a theoretical justification for
clustering algorithms that use the bottom eigenvectors to embed the
vertices into , 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 , and the th smallest eigenvalue
of the normalized Laplacian matrix, denoted . In particular, we show
that in every graph there is a set of size at most which has expansion
at most . This bound is tight, up to constant
factors, for the "noisy hypercube" graphs.Comment: Misc. edits, added reference
Partitioning into Expanders
Let G=(V,E) be an undirected graph, lambda_k be the k-th smallest eigenvalue
of the normalized laplacian matrix of G. There is a basic fact in algebraic
graph theory that lambda_k > 0 if and only if G has at most k-1 connected
components. We prove a robust version of this fact. If lambda_k>0, then for
some 1\leq \ell\leq k-1, V can be {\em partitioned} into l sets P_1,\ldots,P_l
such that each P_i is a low-conductance set in G and induces a high conductance
induced subgraph. In particular, \phi(P_i)=O(l^3\sqrt{\lambda_l}) and
\phi(G[P_i]) >= \lambda_k/k^2).
We make our results algorithmic by designing a simple polynomial time
spectral algorithm to find such partitioning of G with a quadratic loss in the
inside conductance of P_i's. Unlike the recent results on higher order
Cheeger's inequality [LOT12,LRTV12], our algorithmic results do not use higher
order eigenfunctions of G. If there is a sufficiently large gap between
lambda_k and lambda_{k+1}, more precisely, if \lambda_{k+1} >= \poly(k)
lambda_{k}^{1/4} then our algorithm finds a k partitioning of V into sets
P_1,...,P_k such that the induced subgraph G[P_i] has a significantly larger
conductance than the conductance of P_i in G. Such a partitioning may represent
the best k clustering of G. Our algorithm is a simple local search that only
uses the Spectral Partitioning algorithm as a subroutine. We expect to see
further applications of this simple algorithm in clustering applications
High-order Cheeger's inequality on domain
We study the relationship of higher order variational eigenvalues of
p-Laplacian and the higher order Cheeger constants. The asymptotic behavior of
the k-th Cheeger constant is investigated. Using methods of decompostion of the
domain with respect to the eigenfunctions, we obtain the high-order Cheeger's
inequality of p-Laplacian on domain
- …