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Sinkhorn divergence is a measure of dissimilarity between two probabilitymeasures. It is obtained through adding an entropic regularization term toKantorovich's optimal transport problem and can hence be viewed as anentropically regularized Wasserstein distance. Given two discrete probabilityvectors in the $n$-simplex and supported on two bounded spaces in ${\mathbbR}^d$, we present a fast method for computing Sinkhorn divergence when the costmatrix can be decomposed into a $d$-term sum of asymptotically smooth Kroneckerproduct factors. The method combines Sinkhorn's matrix scaling iteration with alow-rank hierarchical representation of the scaling matrices to achieve anear-linear complexity ${\mathcal O}(n \log^3 n)$. This provides a fast andeasy-to-implement algorithm for computing Sinkhorn divergence, enabling itsapplicability to large-scale optimization problems, where the computation ofclassical Wasserstein metric is not feasible. We present a numerical examplerelated to signal processing to demonstrate the applicability of quadraticSinkhorn divergence in comparison with quadratic Wasserstein distance and toverify the accuracy and efficiency of the proposed method.

Hierarchical Low-Rank Approximation of Regularized Wasserstein Distance