Riemannian Geometry Approach

Riemannian geometry offers a powerful framework for analyzing and optimizing machine learning models by moving beyond the limitations of Euclidean space. Current research focuses on applying Riemannian methods to diverse areas, including generative modeling (e.g., using geodesic interpolations and Riemannian score-based models), deep neural network analysis (exploring the geometry of neural network layers and optimization), and efficient optimization algorithms (developing variance-reduced and Hessian-free methods on Riemannian manifolds). This approach enhances model performance and interpretability across various applications, from image classification and trajectory prediction to molecular representation learning and robotic skill acquisition.

Papers

October 24, 2024

A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics
Katharina Friedl, Noémie Jaquier, Jens Lundell, Tamim Asfour, Danica Kragic
Nonlinear Dynamic Reduced Order Lagrangian Method Riemannian Geometry Approach Low Dimensional

May 23, 2024

Metric Flow Matching for Smooth Interpolations on the Data Manifold
Kacper Kapuśniak, Peter Potaptchik, Teodora Reu, Leo Zhang, Alexander Tong, Michael Bronstein, Avishek Joey Bose, Francesco Di Giovanni
Flow Matching Data Manifold Riemannian Geometry Approach Smooth Interpolation Conditional Flow Matching

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

Streamlining in the Riemannian Realm: Efficient Riemannian Optimization with Loopless Variance Reduction
Yury Demidovich, Grigory Malinovsky, Peter Richtárik
Stochastic Optimization Riemannian Optimization Riemannian Geometry Approach Adaptive Variance L SVRG

March 4, 2024

RCoCo: Contrastive Collective Link Prediction across Multiplex Network in Riemannian Space
Li Sun, Mengjie Li, Yong Yang, Xiao Li, Lin Liu, Pengfei Zhang, Haohua Du
Link Prediction Multiplex Network Riemannian Geometry Approach

February 28, 2024

FORML: A Riemannian Hessian-free Method for Meta-learning on Stiefel Manifolds
Hadi Tabealhojeh, Soumava Kumar Roy, Peyman Adibi, Hossein Karshenas
Meta Learning Gradient Based Meta Learning Riemannian Geometry Approach Hessian Free Riemannian Graph Stiefel Manifold

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

June 4, 2023

Riemannian Low-Rank Model Compression for Federated Learning with Over-the-Air Aggregation
Ye Xue, Vincent Lau
Model Compression Low Rank Low Dimensional Manifold Riemannian Geometry Approach Low Rank Compression Low Rank Constraint Air Aggregation

March 27, 2023

Learning Harmonic Molecular Representations on Riemannian Manifold
Yiqun Wang, Yuning Shen, Shi Chen, Lihao Wang, Fei Ye, Hao Zhou
Molecular Representation Protein Ligand Property Prediction 3D Molecule Riemannian Geometry Approach

February 20, 2023

Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning
Wu Lin, Valentin Duruisseaux, Melvin Leok, Frank Nielsen, Mohammad Emtiyaz Khan, Mark Schmidt
Deep Learning Financial Application Riemannian Optimization Riemannian Geometry Approach Covariance Structure Symmetric Positive Definite Matrix Invariant Metric

February 16, 2023

A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold
Yanhong Fei, Xian Wei, Yingjie Liu, Zhengyu Li, Mingsong Chen
Deep Learning Timely Survey Riemannian Manifold Euclidean Space Geometric Optimization Riemannian Geometry Approach

November 23, 2022

BaRe-ESA: A Riemannian Framework for Unregistered Human Body Shapes
Emmanuel Hartman, Emery Pierson, Martin Bauer, Nicolas Charon, Mohamed Daoudi
Riemannian Geometry Approach Riemannian Counterpart Invariant Metric Elastic Shape Analysis

October 27, 2022

Learning Deep Robotic Skills on Riemannian manifolds
Weitao Wang, Matteo Saveriano, Fares J. Abu-Dakka
Skill Learning Riemannian Manifold Riemannian Geometry Approach Pixel Trajectory Motion Manifold

September 19, 2022

Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold
Can Yaras, Peng Wang, Zhihui Zhu, Laura Balzano, Qing Qu
Deep Network Geometric Analysis Neural Collapse Feature Normalization Riemannian Geometry Approach

August 13, 2022

Riemannian accelerated gradient methods via extrapolation
Andi Han, Bamdev Mishra, Pratik Jawanpuria, Junbin Gao
Gradient Method Extrapolation Framework Riemannian Gradient Nesterov Momentum Riemannian Geometry Approach Acceleration Method

July 25, 2022

Riemannian Geometry Approach for Minimizing Distortion and its Applications
Dror Ozeri
Financial Application Affine Transformation Riemannian Metric Riemannian Geometry Approach Fisher Information Metric

March 17, 2022

Visualizing Riemannian data with Rie-SNE
Andri Bergsson, Søren Hauberg
Riemannian Manifold Riemannian Geometry Approach Stochastic Neighbor Riemannian Diffusion

February 6, 2022

Riemannian Score-Based Generative Modelling
Valentin De Bortoli, Emile Mathieu, Michael Hutchinson, James Thornton, Yee Whye Teh, Arnaud Doucet
Generative Model Score Based Generative Riemannian Geometry Approach Riemannian Graph

January 2, 2022

Riemannian Nearest-Regularized Subspace Classification for Polarimetric SAR images
Junfei Shi, Haiyan Jin
Riemannian Geometry Approach Riemannian Counterpart Active Subspace Symmetric Positive Definite Matrix Polarimetric Synthetic Aperture Radar Polarimetric Cue PolSAR Image Classification

December 17, 2021

A singular Riemannian geometry approach to Deep Neural Networks II. Reconstruction of 1-D equivalence classes
Alessandro Benfenati, Alessio Marta
Deep Neural Network Full State Reconstruction Statistical Manifold Riemannian Geometry Approach Neural Mapping Geometric Framework Equivalence Class

Riemannian Geometry Approach

Papers

A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics

Metric Flow Matching for Smooth Interpolations on the Data Manifold

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

Streamlining in the Riemannian Realm: Efficient Riemannian Optimization with Loopless Variance Reduction

RCoCo: Contrastive Collective Link Prediction across Multiplex Network in Riemannian Space

FORML: A Riemannian Hessian-free Method for Meta-learning on Stiefel Manifolds

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

Riemannian Low-Rank Model Compression for Federated Learning with Over-the-Air Aggregation

Learning Harmonic Molecular Representations on Riemannian Manifold

Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning

A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold

BaRe-ESA: A Riemannian Framework for Unregistered Human Body Shapes

Learning Deep Robotic Skills on Riemannian manifolds

Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold

Riemannian accelerated gradient methods via extrapolation

Riemannian Geometry Approach for Minimizing Distortion and its Applications

Visualizing Riemannian data with Rie-SNE

Riemannian Score-Based Generative Modelling

Riemannian Nearest-Regularized Subspace Classification for Polarimetric SAR images

A singular Riemannian geometry approach to Deep Neural Networks II. Reconstruction of 1-D equivalence classes