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

March 12, 2023

Gaussian kernels on non-simply-connected closed Riemannian manifolds are never positive definite
Siran Li
Riemannian Manifold Gaussian Kernel Riemannian Geometry Geodesic Distance

March 5, 2023

CAMEL: Curvature-Augmented Manifold Embedding and Learning
Nan Xu, Yongming Liu
LeArning Abstract Riemannian Geometry Global Embeddings

December 20, 2022

Identifying latent distances with Finslerian geometry
Alison Pouplin, David Eklund, Carl Henrik Ek, Søren Hauberg
Generative Model Riemannian Geometry Riemannian Metric Latent dIstance Model

November 21, 2022

Parametric information geometry with the package Geomstats
Alice Le Brigant, Jules Deschamps, Antoine Collas, Nina Miolane
Riemannian Geometry R Package Information Geometry Probability Density Function ApeRIodic SEmi Parametric

October 22, 2022

Deep Linear Networks for Matrix Completion -- An Infinite Depth Limit
Nadav Cohen, Govind Menon, Zsolt Veraszto
Implicit Regularization Matrix Completion Deep Linear Riemannian Geometry Riemannian Metric Infinite Depth

October 14, 2022

Geometric Reinforcement Learning For Robotic Manipulation
Naseem Alhousani, Matteo Saveriano, Ibrahim Sevinc, Talha Abdulkuddus, Hatice Kose, Fares J. Abu-Dakka
Reinforcement Learning Robotic Manipulation Policy Iteration Traditional Reinforcement Learning Riemannian Geometry Manipulation Skill

September 30, 2022

Riemannian geometry as a unifying theory for robot motion learning and control
Noémie Jaquier, Tamim Asfour
Robotics Domain External Control Riemannian Geometry Unifying Theory Robot Design Muscle Synergy Robot Motion Learning

September 29, 2022

Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective
Yuetian Luo, Nicolas Garcia Trillos
Low Rank Wasserstein Space Riemannian Geometry Landscape Analysis Riemannian Counterpart Positive Semidefinite Burer Monteiro

May 31, 2022

Mario Plays on a Manifold: Generating Functional Content in Latent Space through Differential Geometry
Miguel González-Duque, Rasmus Berg Palm, Søren Hauberg, Sebastian Risi
Latent Space Content Generation Dimensional Manifold Riemannian Geometry Differential Geometry

May 19, 2022

Differentially private Riemannian optimization
Andi Han, Bamdev Mishra, Pratik Jawanpuria, Junbin Gao
Riemannian Gradient Riemannian Geometry Gaussian Mechanism Private Convex Optimization Private Empirical Risk Minimization

March 7, 2022

A Unified Formulation of Geometry-aware Dynamic Movement Primitives
Fares J. Abu-Dakka, Matteo Saveriano, Ville Kyrki
Unified Framework Riemannian Geometry Dynamic Movement Primitive Movement Primitive Parameterized Skill

January 26, 2022

On Sharp Stochastic Zeroth Order Hessian Estimators over Riemannian Manifolds
Tianyu Wang
Hessian Matrix Zeroth Order Linear Estimator Riemannian Manifold Riemannian Geometry

January 17, 2022

Generalization in Supervised Learning Through Riemannian Contraction
Leo Kozachkov, Patrick M. Wensing, Jean-Jacques Slotine
Strong Generalization Gradient Descent Supervised Learning Generalization Bound Convex Loss Riemannian Geometry

December 17, 2021

A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations
Alessandro Benfenati, Alessio Marta
Riemannian Geometry Riemannian Metric Riemannian Geometry Approach Theoretical Foundation Geometric Framework

November 25, 2021

GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition
Xuan Son Nguyen
Action Recognition Skeleton Sequence Gaussian Distribution Riemannian Geometry Skeleton Based Interaction Recognition Symmetric Positive Definite Cholesky Factorization Symmetric Space

November 23, 2021

ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups
Hugo Aguettaz, Erik J. Bekkers, Michaël Defferrard
Geometric Deep Learning Lie Group Equivariant Map Riemannian Geometry Spectral Graph Neural Network Group Equivariance

November 4, 2021

Understanding Riemannian Acceleration via a Proximal Extragradient Framework
Jikai Jin, Suvrit Sra
Gradient Method Riemannian Gradient Riemannian Geometry Riemannian Counterpart Euclidean Geometry