2012/09/12 by Marco Cuturi, Cuturi, Marco
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Optimization Algorithms Research #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Topological and Geometric Data Analysis #math.CO #stat.ML
paper · pdf · doi:10.48550/arxiv.1209.2655
13 pages, 1 figure
arxiv created 2012/09/12 · openalex publication_date 2012/09/12 · arxiv updated 2012/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(R2d) that grows linearly in the complexity of the dimension d, where R2, the total amount of sampled vertices, is a parameter that controls the complexity of the kernel.