  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.
