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In our previous work [arXiv:2202.10042], the complexity of Sinkhorn iterationis reduced from $O(N^2)$ to the optimal $O(N)$ by leveraging the specialstructure of the kernel matrix. In this paper, we explore the special structureof kernel matrices by defining and utilizing the properties of theLower-ColLinear Triangular Matrix (L-CoLT matrix) and Upper-ColLinearTriangular Matrix (U-CoLT matrix). We prove that (1) L/U-CoLT matrix-vectormultiplications can be carried out in $O(N)$ operations; (2) both families ofmatrices are closed under the Hadamard product and matrix scaling. Theseproperties help to alleviate two key difficulties for reducing the complexityof the Inexact Proximal point method (IPOT), and allow us to significantlyreduce the number of iterations to $O(N)$. This yields the Fast Sinkhorn II(FS-2) algorithm for accurate computation of optimal transport with lowalgorithm complexity and fast convergence. Numerical experiments are presentedto show the effectiveness and efficiency of our approach.

Fast Sinkhorn II: Collinear Triangular Matrix and Linear Time Accurate Computation of Optimal Transport