  We prove in this paper that the weighted volume of the set of integral
transportation matrices between two integral histograms r and c of equal sum is
a positive definite kernel of r and c when the set of considered weights forms
a positive definite matrix. The computation of this quantity, despite being the
subject of a significant research effort in algebraic statistics, remains an
intractable challenge for histograms of even modest dimensions. We propose an
alternative kernel which, rather than considering all matrices of the
transportation polytope, only focuses on a sub-sample of its vertices known as
its Northwestern corner solutions. The resulting kernel is positive definite
and can be computed with a number of operations O(R^2d) that grows linearly in
the complexity of the dimension d, where R^2, the total amount of sampled
vertices, is a parameter that controls the complexity of the kernel.
