On metrics robust to noise, perturbations, and changes in projection angle

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.