Year: 2010
Type: article
Abstract: We present a unified framework to study graph kernels, special cases of which include the random \nwalk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; \nMahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time \ncomplexity of kernel computation between unlabeled graphs with n vertices from O(n^6) to O(n^3). \nWe find a spectral decomposition approach even more efficient when computing entire kernel matrices. \nFor labeled graphs we develop conjugate gradient and fixed-point methods that take O(dn^3) \ntime per iteration, where d is the size of the label set. By extending the necessary linear algebra to \nReproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for d-dimensional edge kernels, \nand O(n^4) in the infinite-dimensional case; on sparse graphs these algorithms only take O(n^2) \ntime per iteration in all cases. Experiments on graphs from bioinformatics and other application \ndomains show that these techniques can speed up computation of the kernel by an order of magnitude \nor more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when \nspecialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to \nR-convolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment \nkernel of Fröhlich et al. (2006) yet provably positive semi-definite.
Language: English
Topic: Graph Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2911738047