SVarM: Linear Support Varifold Machines for Classification and Regression on Geometric Data

Shape analysis, or geometric data science, is a field dedicated to building statistical and machine learning methods able to retrieve and analyze the morphological variability in geometric structures. This is a particularly central problem in applications such as computer vision or biomedical imaging where observations often come as segmented curves, surfaces, densities or other types of complex geometric data. Various approaches have been proposed, including Riemannian and elastic shape space models [23, 41, 31, 56], topology based methods [22, 15, 20], and metric/functional matching frameworks [11, 40, 42]. These different methods have proved quite successful in tackling problems such as pairwise comparison, regression, classification or clustering for datasets of shapes. However, with the constant advances in acquisition protocols and the explosion in the size and resolution of datasets that followed, many such methods do not always scale well to recent applications that may involve databases with up to tens of thousands of subjects, each made of potentially hundreds of thousands of vertices. In view of the rapid development of new machine learning paradigms, in particular neural network models, and their impressive achievements in image processing and analysis tasks, one can reasonably expect similar tools to be able to address those challenges on geometric data. Yet, the very particular and intricate nature of shape spaces poses unique challenges when it comes to designing robust neural network models for shape analysis tasks.