Explicit Versus Implicit Graph Feature Maps: A Computational Phase Transition for Walk Kernels

As many real-world data can elegantly be represented as graphs, various graph kernels and methods for computing them have been proposed. Surprisingly, many of the recent graph kernels do not employ the kernel trick anymore but rather compute an explicit feature map and report higher efficiency. So, is there really no benefit of the kernel trick when it comes to graphs? Triggered by this question, we investigate under which conditions it is possible to compute a graph kernel explicitly and for which graph properties this computation is actually more efficient. We give a sufficient condition for R-convolution kernels that enables kernel computation by explicit mapping. We theoretically and experimentally analyze efficiency and flexibility of implicit kernel functions and dot products of explicitly computed feature maps for widely used graph kernels such as random walk kernels, sub graph matching kernels, and shortest-path kernels. For walk kernels we observe a phase transition when comparing runtime with respect to label diversity and walk lengths leading to the conclusion that explicit computations are only favourable for smaller label sets and walk lengths whereas implicit computation is superior for longer walk lengths and data sets with larger label diversity.