  We study the connections between spectral clustering and the problems of
maximum margin clustering, and estimation of the components of level sets of a
density function. Specifically, we obtain bounds on the eigenvectors of graph
Laplacian matrices in terms of the between cluster separation, and within
cluster connectivity. These bounds ensure that the spectral clustering solution
converges to the maximum margin clustering solution as the scaling parameter is
reduced towards zero. The sensitivity of maximum margin clustering solutions to
outlying points is well known, but can be mitigated by first removing such
outliers, and applying maximum margin clustering to the remaining points. If
outliers are identified using an estimate of the underlying probability
density, then the remaining points may be seen as an estimate of a level set of
this density function. We show that such an approach can be used to
consistently estimate the components of the level sets of a density function
under very mild assumptions.
