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

TL;DR

This paper introduces a method for fitting an immersed submanifold to data using flow-based encoders and decoders, leveraging Sussmann's orbit theorem to ensure the reconstructed set lies within the submanifold.

Contribution

It proposes a novel encoder-decoder framework for submanifold fitting that guarantees the reconstructed set is immersed, with theoretical bounds on reconstruction error.

Findings

The method successfully fits submanifolds to data samples.

The approach provides high-probability bounds on excess risk.

It guarantees the reconstructed set is contained within an immersed submanifold.

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.