Topological Constraints on Homeomorphic Auto-Encoding

Understand

When doing representation learning on data that lives on a known non-trivial manifold embedded in high dimensional space, it is natural to desire the encoder to be homeomorphic when restricted to the manifold, so that it is bijective and continuous with a continuous inverse.

• Using topological arguments, we show that when the manifold is non-trivial, the encoder must be globally discontinuous and propose a universal, albeit impractical, construction.
• In addition, we derive necessary constraints which need to be satisfied when designing manifold-specific practical encoders.
• These are used to analyse candidates for a homeomorphic encoder for the manifold of 3D rotations $SO(3)$.