论文标题
在二次瓦斯汀距离的线性化上
On a linearization of quadratic Wasserstein distance
论文作者
论文摘要
本文研究了计算二次瓦斯汀距离线性近似$ W_2 $的问题。特别是，我们计算了负均匀加权索波列夫规范的近似值，该标准与瓦斯汀距离的联系是从一般的蒙格 - 安培方程的经典线性化之后进行的。我们的贡献是三倍。首先，我们提供有关瓦斯汀距离的经典线性化的说明性材料，包括定量误差估计。其次，我们将计算问题减少到解决涉及Witten Laplacian的椭圆边界值问题，该问题是$ H =-Δ+ V $的Schrödinger运算符，并描述了相关的嵌入。第三，对于单位广场上的概率分布$ [0,1]^2 $以$ n \ times n $阵列表示，我们提出了一个快速代码，证明了我们的方法。提出了几个数值示例。
This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance $W_2$. In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving an elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form $H = -Δ+ V$, and describe an associated embedding. Third, for the case of probability distributions on the unit square $[0,1]^2$ represented by $n \times n$ arrays we present a fast code demonstrating our approach. Several numerical examples are presented.
