Topological Constraints on Homeomorphic Auto-Encoding
This addresses a theoretical limitation in representation learning for data on manifolds, which is incremental as it builds on existing topological concepts.
The paper tackles the problem of ensuring homeomorphic encoders for representation learning on known non-trivial manifolds, showing that such encoders must be globally discontinuous and deriving necessary constraints for practical designs, with analysis applied to the manifold of 3D rotations SO(3).
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 $SO(3)$.