Riemannian Geometry

Riemannian geometry provides a powerful mathematical framework for analyzing and modeling data residing in non-Euclidean spaces, addressing limitations of traditional Euclidean approaches. Current research focuses on developing efficient algorithms for manifold learning, including Riemannian extensions of federated learning and gradient descent methods, and applying these techniques to diverse data types such as EEG signals, covariance matrices, and graph data. This work is significant for improving the accuracy and interpretability of machine learning models in various fields, from brain-computer interfaces to robotics and image analysis, by leveraging the intrinsic geometric structure of the data.

Papers

October 2, 2024

Score-based pullback Riemannian geometry
Willem Diepeveen, Georgios Batzolis, Zakhar Shumaylov, Carola-Bibiane Schönlieb
Score Based Riemannian Geometry Riemannian Graph Riemannian Classifier

September 23, 2024

Isometric Immersion Learning with Riemannian Geometry
Zihao Chen, Wenyong Wang, Yu Xiang
Manifold Learning Riemannian Geometry Geometry Prior Isometric Mapping Isometric Representation

September 18, 2024

Cartan moving frames and the data manifolds
Eliot Tron, Rita Fioresi, Nicolas Couellan, Stéphane Puechmorel
Neural Network Geometric Analysis Indoor Wall Frame Installation Gaussian Curvature Riemannian Geometry Data Manifold

September 11, 2024

Riemannian Federated Learning via Averaging Gradient Stream
Zhenwei Huang, Wen Huang, Pratik Jawanpuria, Bamdev Mishra
Federated Averaging Riemannian Geometry FedAvg Converges Riemannian Stochastic

July 19, 2024

Riemannian Geometry-Based EEG Approaches: A Literature Review
Imad Eddine Tibermacine, Samuele Russo, Ahmed Tibermacine, Abdelaziz Rabehi, Bachir Nail, Kamel Kadri, Christian Napoli
Literature Review Riemannian Geometry Riemannian Classifier Brain Computer Interface Competition

July 15, 2024

Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry
Ziheng Chen, Yue Song, Xiao-Jun Wu, Gaowen Liu, Nicu Sebe
Camera Lens Covariance Matrix Riemannian Geometry Euclidean Neural Network Riemannian Classifier Global Covariance Pooling

July 2, 2024

A mathematical framework of intelligence and consciousness based on Riemannian Geometry
Meng Lu
Cognitive Science Cognitive Intelligence Human Consciousness Riemannian Geometry Geometric Understanding

May 22, 2024

Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold
Sixin Zhang
Riemannian Geometry Local Convergence Wasserstein Gan Equilibrium State Stackelberg Equilibrium Differentiable Game

May 16, 2024

Manifold Integrated Gradients: Riemannian Geometry for Feature Attribution
Eslam Zaher, Maciej Trzaskowski, Quan Nguyen, Fred Roosta
Feature Attribution Feature Attribution Method Riemannian Geometry Integrated Gradient Path Modeling Attribution Attack

April 15, 2024

Federated Learning on Riemannian Manifolds with Differential Privacy
Zhenwei Huang, Wen Huang, Pratik Jawanpuria, Bamdev Mishra
Differential Privacy Private Federated Learning Riemannian Manifold Riemannian Geometry Convergence Property

April 9, 2024

A singular Riemannian Geometry Approach to Deep Neural Networks III. Piecewise Differentiable Layers and Random Walks on $n$-dimensional Classes
Alessandro Benfenati, Alessio Marta
Neural Network Deep Neural Network Random Walk Riemannian Geometry Differentiable Neural Riemannian Geometry Approach Recursive Neural Network Differentiable Activation Function

March 11, 2024

Pulling back symmetric Riemannian geometry for data analysis
Willem Diepeveen
Riemannian Geometry Symmetric Space

January 25, 2024

Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs
Martin Hanik, Gabriele Steidl, Christoph von Tycowicz
Convolutional Neural Network Multi Layer Riemannian Geometry Graph Diffusion Latent Manifold Neural Network Layer Manifold Graph

November 27, 2023

Riemannian Prediction of Anatomical Diagnoses in Congenital Heart Disease based on 12-lead ECGs
Muhammet Alkan, Gruschen Veldtman, Fani Deligianni
Anatomical Structure Riemannian Geometry Lead ECG Riemannian Geometry Approach Covariance Neural Network

October 11, 2023

Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning
Noémie Jaquier, Leonel Rozo, Tamim Asfour
General Analysis Robot Learning Rigid Body Riemannian Geometry Manipulability Maximization Plausible Clarification Tangent Space

September 14, 2023

Structure-Preserving Transformers for Sequences of SPD Matrices
Mathieu Seraphim, Alexis Lechervy, Florian Yger, Luc Brun, Olivier Etard
Transformer Megatron Decepticons Sequence of Sequence Riemannian Geometry Non Euclidean Transformer Attention Symmetric Positive Definite Symmetric Positive Definite Matrix Multivariate Biosignals

July 28, 2023

On the Design of Region-Avoiding Metrics for Collision-Safe Motion Generation on Riemannian Manifolds
Holger Klein, Noémie Jaquier, Andre Meixner, Tamim Asfour
Product Design Collision Free Metric Library Riemannian Manifold Riemannian Geometry Riemannian Metric

July 24, 2023

Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry
Yong-Hyun Park, Mingi Kwon, Jaewoong Choi, Junghyo Jo, Youngjung Uh
Diffusion Model Latent Space Camera Lens Latent Structure Riemannian Geometry Diffusion Step

July 10, 2023

On the curvature of the loss landscape
Alison Pouplin, Hrittik Roy, Sidak Pal Singh, Georgios Arvanitidis
Deep Network Loss Landscape Gaussian Curvature Riemannian Geometry Differential Geometry

April 18, 2023

Implicit representation priors meet Riemannian geometry for Bayesian robotic grasping
Norman Marlier, Julien Gustin, Olivier Brüls, Gilles Louppe
Robotic Grasping Implicit Representation Riemannian Geometry Grasp Pose Implicit Prior Scene Prior

Riemannian Geometry

Papers

Score-based pullback Riemannian geometry

Isometric Immersion Learning with Riemannian Geometry

Cartan moving frames and the data manifolds

Riemannian Federated Learning via Averaging Gradient Stream

Riemannian Geometry-Based EEG Approaches: A Literature Review

Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry

A mathematical framework of intelligence and consciousness based on Riemannian Geometry

Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold

Manifold Integrated Gradients: Riemannian Geometry for Feature Attribution

Federated Learning on Riemannian Manifolds with Differential Privacy

A singular Riemannian Geometry Approach to Deep Neural Networks III. Piecewise Differentiable Layers and Random Walks on $n$-dimensional Classes

Pulling back symmetric Riemannian geometry for data analysis

Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs

Riemannian Prediction of Anatomical Diagnoses in Congenital Heart Disease based on 12-lead ECGs

Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning

Structure-Preserving Transformers for Sequences of SPD Matrices

On the Design of Region-Avoiding Metrics for Collision-Safe Motion Generation on Riemannian Manifolds

Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry

On the curvature of the loss landscape

Implicit representation priors meet Riemannian geometry for Bayesian robotic grasping