Fitting an immersed submanifold to data via Sussmann’s orbit theorem

This paper describes an approach for fitting an immersed submanifold of a finite-dimensional Euclidean space to random samples. The reconstruction mapping from the ambient space to the desired submanifold is implemented as a composition of an encoder that maps each point to a tuple of (positive or negative) times and a decoder given by a composition of flows along finitely many vector fields starting from a fixed initial point. The encoder supplies the times for the flows. The encoder–decoder map is obtained by empirical risk minimization, and a high-probability bound is given on the excess risk relative to the minimum expected reconstruction error over a given class of encoder–decoder maps. The proposed approach makes fundamental use of Sussmann’s orbit theorem, which guarantees that the image of the reconstruction map is indeed contained in an immersed submanifold.