We study 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. 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 perturbations and that the distance between one-dimensional tomographic projections of a two-dimensional function is bounded by the size of the difference in projection angles. We also establish error bounds for approximating the metric from finite samples, and prove that these approximations are robust to additive Gaussian noise. The results are illustrated in numerical experiments.

翻译：摘要：我们研究了单变量函数间的一族距离。这些距离是积分概率度量的实例，且此前已被用于概率测度的比较。其特例包括推土机距离和柯尔莫哥洛夫度量。我们探究了它们对一般信号的性质，证明了它们对一大类扰动具有鲁棒性，并且二维函数的一维断层扫描投影之间的距离受投影角度差异大小的约束。我们还建立了基于有限样本近似该度量的误差界，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。