Topological Constraints on Homeomorphic Auto-Encoding

Pim de Haan; Luca Falorsi

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arXiv · 1812.10783

Topological Constraints on Homeomorphic Auto-Encoding

Pim de Haan·Luca Falorsi

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Abstract

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 $SO(3)$.

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Pim de Haan, Luca Falorsi. 2018-12-27. Topological Constraints on Homeomorphic Auto-Encoding. https://arxiv.org/abs/1812.10783

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