47 research outputs found
Consistency of spectral clustering in stochastic block models
We analyze the performance of spectral clustering for community extraction in
stochastic block models. We show that, under mild conditions, spectral
clustering applied to the adjacency matrix of the network can consistently
recover hidden communities even when the order of the maximum expected degree
is as small as , with the number of nodes. This result applies to
some popular polynomial time spectral clustering algorithms and is further
extended to degree corrected stochastic block models using a spherical
-median spectral clustering method. A key component of our analysis is a
combinatorial bound on the spectrum of binary random matrices, which is sharper
than the conventional matrix Bernstein inequality and may be of independent
interest.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1274 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
How to Round Subspaces: A New Spectral Clustering Algorithm
A basic problem in spectral clustering is the following. If a solution
obtained from the spectral relaxation is close to an integral solution, is it
possible to find this integral solution even though they might be in completely
different basis? In this paper, we propose a new spectral clustering algorithm.
It can recover a -partition such that the subspace corresponding to the span
of its indicator vectors is close to the original subspace in
spectral norm with being the minimum possible ( always).
Moreover our algorithm does not impose any restriction on the cluster sizes.
Previously, no algorithm was known which could find a -partition closer than
.
We present two applications for our algorithm. First one finds a disjoint
union of bounded degree expanders which approximate a given graph in spectral
norm. The second one is for approximating the sparsest -partition in a graph
where each cluster have expansion at most provided where is the eigenvalue of
Laplacian matrix. This significantly improves upon the previous algorithms,
which required .Comment: Appeared in SODA 201