Generalized vec trick for fast learning of pairwise kernel models

Pairwise learning corresponds to the supervised learning setting where the goal is to make predictions for pairs of objects. Prominent applications include predicting drug-target or protein-protein interactions, or customer-product preferences. Several kernel functions have been proposed for incorporating prior knowledge about the relationship between the objects, when training kernel based learning methods. However, the number of training pairs n is often very large, making O(n^2) cost of constructing the pairwise kernel matrix infeasible. If each training pair x= (d,t) consists of drug d and target t, let m and q denote the number of unique drugs and targets appearing in the training pairs. In many real-world applications m,q << n, which can be used to develop computational shortcuts. Recently, a O(nm+nq) time algorithm we refer to as the generalized vec trick was introduced for training kernel methods with the Kronecker kernel. In this work, we show that a large class of pairwise kernels can be expressed as a sum of product matrices, which generalizes the result to the most commonly used pairwise kernels. This includes symmetric and anti-symmetric, metric-learning, Cartesian, ranking, as well as linear, polynomial and Gaussian kernels. In the experiments, we demonstrate how the introduced approach allows scaling pairwise kernels to much larger data sets than previously feasible, and compare the kernels on a number of biological interaction prediction tasks.