Spectral Partitioning with Inde nite Kernels using the Nystr om Extension

Fowlkes et al recently introduced an approximation to the Normalized Cut NCut grouping algorithm based on random subsampling and the Nystr om extension As presented their method is restricted to the case where W the weighted adjacency matrix is positive de nite Although many common measures of image similarity i e kernels are positive de nite a popular example being Gaussian weighted distance there are important cases that are not In this work we present a modi cation to Nystr om NCut that does not require W to be positive de nite The modi cation only a ects the orthogonalization step and in doing so it necessitates one additional O m operation where m is the number of random samples used in the approximation As such it is of interest to know which kernels are positive de nite and which are inde nite In addressing this issue we further develop connections between NCut and related methods in the kernel machines literature We provide a proof that the Gaussian weighted chi squared kernel is positive de nite which has thus far only been conjectured We also explore the performance of the approximation algorithm on a variety of grouping cues including contour color and texture