论文标题
正则遗嘱式距离分层低级近似
Hierarchical Low-Rank Approximation of Regularized Wasserstein Distance
论文作者
论文摘要
sindhorn差异是两种概率度量之间差异的量度。它是通过将熵正则术语添加到Kantorovich的最佳运输问题中获得的，因此可以将其视为熵正规化的Wasserstein距离。鉴于$ n $ simplex中的两个离散的概率向量，并在$ {\ Mathbb r}^d $中的两个有界空间上支撑，我们提出了一种快速的方法，用于计算sindhorn差异时，当成本矩阵可以分解为$ d $ d $ derm-d $ derm-d $ - ymptotal sum-derm-d $ - ymptotically sum-term y-d $ a的非额外平滑kronecker kronecker ruatsex froduct froduct actuffect。该方法将Sindhorn的矩阵缩放率与缩放矩阵的低级别层次表示形式相结合，以实现接近线性的复杂性$ {\ Mathcal O}（\ n \ log^3 n）$。这提供了一种快速，易于实施的算法，用于计算sindhorn差异，使其适用于大规模优化问题，而经典的Wasserstein Metric的计算是不可行的。我们提出了与信号处理相关的数值示例，以证明与二次瓦斯坦距离距离相比，二次凹痕差异的适用性，并验证所提出方法的准确性和效率。
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.
