正则化Wasserstein距离的分层低秩逼近
Hierarchical Low-Rank Approximation of Regularized Wasserstein Distance
摘要：
Sinkhorn散度是两个概率测度之间相异性的度量。它是通过在Kantorovich的最优输运问题中加入一个熵正则化项而得到的，因此可以看作是一个熵正则化的Wasserstein距离。给定$n$ -单纯形中的两个离散概率向量，并支撑在${\mathbb R}^d$中的两个有界空间上，当代价矩阵可分解为渐近光滑Kronecker乘积因子的$d$ -项和时，我们给出了计算Sinkhorn发散的一个快速方法.该方法结合Sinkhorn的矩阵缩放迭代与低秩层次表示的缩放矩阵，以实现近线性的复杂度${\mathcal O}(n \log^3 n)$。这为计算Sinkhorn散度提供了一种快速且易于实现的算法，使其适用于大规模最优化问题，其中经典Wasserstein度量的计算是不可行的。我们提出了一个与信号处理相关的数值例子，以证明二次Sinkhorn发散的适用性，与二次Wasserstein距离相比，并验证所提出的方法的准确性和效率。
英文摘要：
Sinkhorn divergence is a measure of dissimilarity between two probability measures. It is obtained through adding an entropic regularization term to Kantorovich's optimal transport problem and can hence be viewed as an entropically regularized Wasserstein distance. Given two discrete probability vectors in the $n$-simplex and supported on two bounded spaces in ${\mathbb R}^d$, we present a fast method for computing Sinkhorn divergence when the cost matrix can be decomposed into a $d$-term sum of asymptotically smooth Kronecker product factors. The method combines Sinkhorn's matrix scaling iteration with a low-rank hierarchical representation of the scaling matrices to achieve a near-linear complexity ${\mathcal O}(n \log^3 n)$. This provides a fast and easy-to-implement algorithm for computing Sinkhorn divergence, enabling its applicability to large-scale optimization problems, where the computation of classical Wasserstein metric is not feasible. We present a numerical example related to signal processing to demonstrate the applicability of quadratic Sinkhorn divergence in comparison with quadratic Wasserstein distance and to verify the accuracy and efficiency of the proposed method.
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