Autoencoding Dynamics: Topological Limitations and Capabilities
This work addresses theoretical limitations in autoencoder design for researchers in machine learning and dynamical systems, but it appears incremental as it builds on existing topological concepts.
The paper investigates the topological constraints and possibilities for autoencoders when mapping data manifolds to latent spaces, and explores their application to dynamical systems with invariant manifolds.
Given a "data manifold" $M\subset \mathbb{R}^n$ and "latent space" $\mathbb{R}^\ell$, an autoencoder is a pair of continuous maps consisting of an "encoder" $E\colon \mathbb{R}^n\to \mathbb{R}^\ell$ and "decoder" $D\colon \mathbb{R}^\ell\to \mathbb{R}^n$ such that the "round trip" map $D\circ E$ is as close as possible to the identity map $\mbox{id}_M$ on $M$. We present various topological limitations and capabilites inherent to the search for an autoencoder, and describe capabilities for autoencoding dynamical systems having $M$ as an invariant manifold.