  The success of kernel methods has initiated the design of novel positive
semidefinite functions, in particular for structured data. A leading design
paradigm for this is the convolution kernel, which decomposes structured
objects into their parts and sums over all pairs of parts. Assignment kernels,
in contrast, are obtained from an optimal bijection between parts, which can
provide a more valid notion of similarity. In general however, optimal
assignments yield indefinite functions, which complicates their use in kernel
methods. We characterize a class of base kernels used to compare parts that
guarantees positive semidefinite optimal assignment kernels. These base kernels
give rise to hierarchies from which the optimal assignment kernels are computed
in linear time by histogram intersection. We apply these results by developing
the Weisfeiler-Lehman optimal assignment kernel for graphs. It provides high
classification accuracy on widely-used benchmark data sets improving over the
original Weisfeiler-Lehman kernel.
