Kernel methods for graphs

The computational cost of applying kernel methods to graphs is typically dominated by the cost of computing the kernel matrix. This contrasts with the more common setting where the kernel between two instances can be computed individually and cheaply. This holds for example in text classification and other domains where the instances are sparse or dense vectors in some Euclidean space. Several toolkits for this case have been developed and are widely used. The approaches to cope with the size of the graphs vary and real-world applications so far have suered from a lack of clear guidance which technique to use in which situation. In this work we survey possible techniques and apply them in one setting. In particular we concentrate on regularised least squares regression and support vector machines with powerseries kernels K = P1 i iB i where B is a sparse matrix representing the structure of the graph such as the (normalised) Laplacian or the adjacency matrix. Using of-the-shelf iterative solvers we obtain highly scalable implementations for these settings that exploit the sparsity of the graphs. We distinguish techniques for general kernel matrices of power series form and methods that are only applicable to kernel matrices defined by geometric power series ( i = i). While the first kernel matrix to be proposed—the "dif- fusion kernel"—was defined using the exponential powerseries ( i = i/i!), the geometric powerseries appears more popular nowadays. The main dierence is that for a closed form computation of the general power series we need to com- pute the eigenvalues and apply an anti-monotone transformation. For geometric powerseries, we can make use of the fact that the closed form is the inverse of a sparse matrix, e.g., K = J 1 = (L + I) 1. The iterative methods we propose have time complexity linear in the number of edges of the graph on (a particular kind of) random graphs constructed with a fixed number of dense regions.