Faster computation of the direct product kernel for graph classification

The direct product kernel, introduced by Gartner et al. for graph classification, is based on defining a feature for every possible label sequence in a labelled graph and counting how many label sequences in two given graphs are identical. Although the direct product kernel has achieved promising results in terms of accuracy, the kernel computation is not feasible for large graphs. This is because computing the direct product kernel for two graphs is essentially computing either the inverse of or by diagonalizing the adjacency matrix of the direct product of these two graphs. For two graphs with adjacency matrices of sizes m and n, the adjacency matrix of their direct product graph can be of size mn in the worst case. As both matrix inversion or matrix diagonalizing in the general case is O(n³), computing the direct product kernel is O((mn)³). Our survey of data sets in graph classification indicates that most graphs have adjacency matrices of sizes in the order of hundreds which often leads to adjacency matrices of direct product graphs (of two graphs) having sizes in the order of thousands. In this work we show how the direct product kernel can be computed in O((m + n)³). The key insight behind our result is that the language of label sequences in a labeled graph is a regular language and that regular languages are closed under union and intersection.