Monge-Kantorovich quantiles and ranks for image data

Abstract: This paper defines quantiles, ranks and statistical depths for image data by leveraging ideas from measure transportation. The first step is to embed a distribution of images in a tangent space, with the framework of linear optimal transport. Therein, Monge-Kantorovich quantiles are shown to provide a meaningful ordering of image data, with outward images having unusual shapes. Numerical experiments showcase the relevance of the proposed procedure, for descriptive analysis, outlier detection or statistical testing.