Turing approximations, toric isometric embeddings & manifold convolutions

TL;DR

This paper introduces a theoretical framework for defining convolutions on manifolds by embedding them into tori, enabling global convolution operations on complex topologies, which addresses computational intractability issues of local methods.

Contribution

It proposes a novel approach combining extrinsic and intrinsic methods via toric isometric embeddings to define manifold convolutions for arbitrary topologies.

Findings

Defines a convolution operator for manifolds of any topology and dimension.

Identifies geometric and topological conditions affecting local convolution definitions.

Highlights the importance of toric embeddings for global convolution in finite metric spaces.

Abstract

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.