Positivity and Transportation
This addresses a computational bottleneck in algebraic statistics for researchers working with transportation matrices and histograms, though it is an incremental improvement over existing methods.
The paper tackles the computational intractability of computing weighted volumes of transportation matrices between histograms by proposing an alternative kernel that focuses only on Northwestern corner solutions. This kernel is positive definite and can be computed with O(R^2d) operations, where R^2 controls complexity and d is the dimension.
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.