We study the properties of a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures on the line; special cases include the Earth Mover's Distance and the Kolmogorov Metric. We examine their properties for general signals, proving that they are robust to a broad class of deformations. We also establish corresponding robustness results for the induced sliced distances between multivariate functions. Finally, we establish error bounds for approximating the univariate metrics from finite samples, and prove that these approximations are robust to additive Gaussian noise. The results are illustrated in numerical experiments, which include comparisons with Wasserstein distances.

翻译：我们研究单变量函数间一族距离的性质。这些距离是积分概率度量的特例，已被用于比较直线上的概率测度；特殊情形包括推土机距离和柯尔莫哥洛夫度量。我们考察它们对一般信号的性质，证明其对于一大类形变具有鲁棒性。我们还建立了多维函数诱导切片距离的相应鲁棒性结果。最后，我们建立了从有限样本近似单变量度量的误差界，并证明这些近似对加性高斯噪声具有鲁棒性。数值实验验证了这些结果，其中包含与Wasserstein距离的对比。