Volume 2352 of the series Lecture Notes in Computer Science pp 531542
Spectral Partitioning with Indefinite Kernels Using the Nyström Extension
 Serge BelongieAffiliated withUniversity of California
 , Charless FowlkesAffiliated withUniversity of California, Berkeley
 , Fan ChungAffiliated withUniversity of California
 , Jitendra MalikAffiliated withUniversity of California, Berkeley
Abstract
Fowlkes et al. [7] recently introduced an approximation to the Normalized Cut (NCut) grouping algorithm [18] based on random subsampling and the Nyström extension. As presented, their method is restricted to the case where W, the weighted adjacency matrix, is positive definite. Although many common measures of image similarity (i.e. kernels) are positive definite, a popular example being Gaussianweighted distance, there are important cases that are not. In this work, we present a modification to NyströmNCut that does not require W to be positive definite. The modification only affects the orthogonalization step, and in doing so it necessitates one additional O(m ^{3}) 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 definite and which are indefinite. In addressing this issue, we further develop connections between NCut and related methods in the kernel machines literature. We provide a proof that the Gaussianweighted chisquared kernel is positive definite, 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.
 Title
 Spectral Partitioning with Indefinite Kernels Using the Nyström Extension
 Book Title
 Computer Vision — ECCV 2002
 Book Subtitle
 7th European Conference on Computer Vision Copenhagen, Denmark, May 28–31, 2002 Proceedings, Part III
 Pages
 pp 531542
 Copyright
 2002
 DOI
 10.1007/3540479775_35
 Print ISBN
 9783540437468
 Online ISBN
 9783540479772
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 2352
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Anders Heyden ^{(4)}
 Gunnar Sparr ^{(4)}
 Mads Nielsen ^{(5)}
 Peter Johansen ^{(6)}
 Editor Affiliations

 4. Centre for Mathematical Sciences, Lund University
 5. The IT University of Copenhagen
 6. University of Copenhagen
 Authors

 Serge Belongie ^{(7)}
 Charless Fowlkes ^{(8)}
 Fan Chung ^{(7)}
 Jitendra Malik ^{(8)}
 Author Affiliations

 7. University of California, San Diego, La Jolla, CA, 92093, USA
 8. University of California, Berkeley, Berkeley, CA, 94720, USA
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