Fitting an immersed submanifold to data via Sussmann’s orbit theorem
- Resource Type
- Conference
- Authors
- Hanson, Joshua; Raginsky, Maxim
- Source
- 2022 IEEE 61st Conference on Decision and Control (CDC) Decision and Control (CDC), 2022 IEEE 61st Conference on. :5323-5328 Dec, 2022
- Subject
- Robotics and Control Systems
Space vehicles
Neural networks
Fitting
Orbits
Mathematical models
Manifold learning
Risk management
- Language
- ISSN
- 2576-2370
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.