Sub-Riemannian landmark matching and its interpretation as residual neural networks

Journal article, 2025

The problem of finding a time-dependent vector field which warps an initial set of points to a target set is common in shape analysis. It is an example of a problem in the diffeomorphic shape matching regime, and can be thought of as a spatial discretization of diffeomorphic image matching. In this paper, we consider landmark matching modified by restricting the set of available vector fields in the sense that vector fields are parametrized by a set of controls. We determine the geometric setting of the problem, referred to as sub-Riemannian landmark matching, and derive the equations of motion for the controls. We provide two computational algorithms and demonstrate them in numerical examples. In particular, the experiments highlight the importance of the regularization term. A strong motivation is that sub-Riemannian landmark matching have connections with neural networks, in particular the interpretation of residual neural networks as time discretizations of continuous control problems. It allows shape analysis practitioners to think about neural networks in terms of diffeomorphic landmark matching, thereby providing a bridge between the two fields.