Measuring dissimilarity with diffeomorphism invariance

One of the overarching goals of most machine learning algorithms is to generalize to unseen data. Ensuring and quantifying generalization is of course challenging, especially in the high-dimensional setting. One way of reducing the hardness of a learning problem is to study the invariances that may exist with respect to the distribution of data, effectively reducing its dimension. Handling invariances in data has attracted considerable attention over time in machine learning and applied mathematics more broadly. Two notable examples are image registration [De Castro and Morandi, 1987, Reddy and Chatterji, 1996] and time series alignment [Sakoe and Chiba, 1978, Cuturi and Blondel, 2017, Vayer et al., 2020, Blondel et al., 2021, Cohen et al., 2021]. In practice, data augmentation is a central tool in the machine learning practitioner's toolbox. In computer vision for instance, images are randomly cropped, color spaces are changed, artifacts are added.