# Turing approximations, toric isometric embeddings & manifold convolutions

arXiv: https://arxiv.org/abs/2110.02279 · HTML: https://mlsanity.com/paper/2110.02279/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 61 references that are in the map.
- First posted on arXiv 5 Oct 2021.

## Builds on (most cited first, 17 of 17)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [VoxelNet: End-to-End Learning for Point Cloud Based 3D Object Detection](https://mlsanity.com/paper/1711.06396/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [3D Semantic Segmentation with Submanifold Sparse Convolutional Networks](https://mlsanity.com/paper/1711.10275/) (2017)
- [OctNet: Learning Deep 3D Representations at High Resolutions](https://mlsanity.com/paper/1611.05009/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [Multi-directional Geodesic Neural Networks via Equivariant Convolution](https://mlsanity.com/paper/1810.02303/) (2018)
- [Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis](https://mlsanity.com/paper/1709.03615/) (2017)
- [Parallel Transport Convolution: A New Tool for Convolutional Neural Networks on Manifolds](https://mlsanity.com/paper/1805.07857/) (2018)

## Built on by (most cited first, 1 of 1)

- [Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders](https://mlsanity.com/paper/2208.10570/) (2022)

## Abstract (arXiv)

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.