Turing approximations, toric isometric embeddings & manifold convolutions
This work addresses a foundational problem in deep learning for researchers and practitioners working with manifold data, but it appears incremental as it builds on existing theoretical concepts without demonstrating practical applications or empirical results.
The authors tackled the problem of defining convolution operators on manifolds with arbitrary topology and dimension by proposing a theoretical framework that combines extrinsic and intrinsic approaches through isometric embeddings into tori, resulting in a global definition of convolution that addresses computational intractability issues in local methods.
Convolutions are fundamental elements in deep learning architectures. Here, we present a theoretical framework for combining extrinsic and intrinsic approaches to manifold convolution through isometric embeddings into tori. In this way, we define a convolution operator for a manifold of arbitrary topology and dimension. We also explain geometric and topological conditions that make some local definitions of convolutions which rely on translating filters along geodesic paths on a manifold, computationally intractable. A result of Alan Turing from 1938 underscores the need for such a toric isometric embedding approach to achieve a global definition of convolution on computable, finite metric space approximations to a smooth manifold.