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

Abstract: 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.