Tangent Bundle

Tangent bundles, mathematical structures associating a vector space to each point of a manifold, are increasingly used in machine learning to represent and process data with complex geometric or topological properties. Current research focuses on developing novel neural network architectures, such as bundle neural networks and tangent bundle convolutional networks, that leverage the tangent bundle framework for tasks like graph data processing, image transformation, and dimensionality reduction. These approaches aim to overcome limitations of traditional methods by incorporating geometric information, improving model expressivity, and enabling more accurate and efficient learning on non-Euclidean data. The resulting advancements have significant implications for various fields, including computer vision, graph analysis, and data visualization.

Papers

May 24, 2024

Bundle Neural Networks for message diffusion on graphs
Jacob Bamberger, Federico Barbero, Xiaowen Dong, Michael Bronstein
Graph Drawing Information Diffusion Tangent Bundle

October 26, 2023

Optimization dependent generalization bound for ReLU networks based on sensitivity in the tangent bundle
Dániel Rácz, Mihály Petreczky, András Csertán, Bálint Daróczy
Deep Learning Deep Neural Network Optimization Purpose Gradient Descent Generalization Error ReLU Network Network Sensitivity Tangent Bundle

July 24, 2023

Entropy Transformer Networks: A Learning Approach via Tangent Bundle Data Manifold
Pourya Shamsolmoali, Masoumeh Zareapoor
CNN Architecture Convolution Operation Image Transformation Spatial Transformer Network Learning Approach Interpolation Method Transformer Based Entropy Model Tangent Bundle

March 20, 2023

Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back
Claudio Battiloro, Zhiyang Wang, Hans Riess, Paolo Di Lorenzo, Alejandro Ribeiro
Class Manifold Sheaf Diffusion Tangent Bundle

October 26, 2022

Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back
Claudio Battiloro, Zhiyang Wang, Hans Riess, Paolo Di Lorenzo, Alejandro Ribeiro
Neural Network Class Manifold Sheaf Diffusion Tangent Bundle

June 13, 2022

FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles
Luis Scoccola, Jose A. Perea
Dimensionality Reduction Arbitrary Topology Short Fiber Tangent Bundle

December 17, 2021

Complex Functional Maps : a Conformal Link Between Tangent Bundles
Nicolas Donati, Etienne Corman, Simone Melzi, Maks Ovsjanikov
Vector Field Functional Map Conformal Method Conformal Mapping Orientation Representation Tangent Bundle

Tangent Bundle

Papers

Bundle Neural Networks for message diffusion on graphs

Optimization dependent generalization bound for ReLU networks based on sensitivity in the tangent bundle

Entropy Transformer Networks: A Learning Approach via Tangent Bundle Data Manifold

Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back

Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back

FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles

Complex Functional Maps : a Conformal Link Between Tangent Bundles