Cartesian Kernel: An Efficient Alternative to the Pairwise Kernel

Hisashi KASHIMA  Satoshi OYAMA  Yoshihiro YAMANISHI  Koji TSUDA  

IEICE TRANSACTIONS on Information and Systems   Vol.E93-D   No.10   pp.2672-2679
Publication Date: 2010/10/01
Online ISSN: 1745-1361
DOI: 10.1587/transinf.E93.D.2672
Print ISSN: 0916-8532
Type of Manuscript: Special Section PAPER (Special Section on Data Mining and Statistical Science)
kernel methods,  pairwise kernels,  link prediction,  

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Pairwise classification has many applications including network prediction, entity resolution, and collaborative filtering. The pairwise kernel has been proposed for those purposes by several research groups independently, and has been used successfully in several fields. In this paper, we propose an efficient alternative which we call a Cartesian kernel. While the existing pairwise kernel (which we refer to as the Kronecker kernel) can be interpreted as the weighted adjacency matrix of the Kronecker product graph of two graphs, the Cartesian kernel can be interpreted as that of the Cartesian graph, which is more sparse than the Kronecker product graph. We discuss the generalization bounds of the two pairwise kernels by using eigenvalue analysis of the kernel matrices. Also, we consider the N-wise extensions of the two pairwise kernels. Experimental results show the Cartesian kernel is much faster than the Kronecker kernel, and at the same time, competitive with the Kronecker kernel in predictive performance.