Autoencoding Dynamics: Topological Limitations and Capabilities

Abstract

Given a “data manifold” M ⊂ Rⁿ and “latent space” R<superscript>ℓ</superscript>, an autoencoder is a pair ofcontinuous maps consisting of an “encoder” E : Rⁿ → R<superscript>ℓ</superscript>and “decoder” D : R<superscript>ℓ</superscript> → Rⁿ such that the “round trip” map D ◦ E|ₘ is as close as possible to the identity map idₘ 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.