Using extensions of linear algebra concepts to Reproducing Kernel Hilbert Spaces (RKHS), we define a unifying framework for random walk kernels on graphs. Reduction to a Sylvester equation allows us to compute many of these kernels in O(n3) worst-case time. This includes kernels whose previous worst-case time complexity was O(n6), such as the geometric kernels of Gartner et al.  and the marginal graph kernels of Kashima et al. . Our algebra in RKHS allow us to exploit sparsity in directed and undirected graphs more effectively than previous methods, yielding sub-cubic computational complexity when combined with conjugate gradient solvers or fixed-point iterations. Experiments on graphs from bioinformatics and other application domains show that our algorithms are often more than 1000 times faster than existing approaches.
Stephen J. Wright,et al.
Hans-Peter Kriegel,et al.
Protein function prediction via graph kernels
Gene H. Golub,et al.
H. Kashima,et al.
Kernels for graphs
Thomas Gärtner,et al.
On Graph Kernels: Hardness Results and Efficient Alternatives
The ubiquitous Kronecker product
Alan J. Laub,et al.
Solution of the Sylvester matrix equation AXBT + CXDT = E
Frank Harary,et al.