Geodesic Metric Space

Geodesic metric spaces, which generalize Euclidean spaces to curved manifolds, are becoming increasingly important in various fields. Current research focuses on developing algorithms for optimization and machine learning tasks within these spaces, including Bayesian optimization, clustering, and min-max problems, often employing first-order methods and geometric metrics like density and coverage for improved performance and robustness. This work addresses challenges posed by the curvature and non-Euclidean geometry, leading to more accurate and efficient solutions for applications ranging from robotics and 3D modeling to adversarial example detection. The development of efficient algorithms and appropriate metrics within geodesic metric spaces is crucial for advancing these diverse applications.

Papers

January 30, 2024

A Group Theoretic Metric for Robot State Estimation Leveraging Chebyshev Interpolation
Varun Agrawal, Frank Dellaert
Ground Truth State Estimation Chebyshev Davidson Algorithm Trajectory Estimation Geodesic Metric Space

January 3, 2024

Beyond Regrets: Geometric Metrics for Bayesian Optimization
Jungtaek Kim
Bayesian Optimization Global Optimum Geodesic Metric Space

June 27, 2023

Toward Mesh-Invariant 3D Generative Deep Learning with Geometric Measures
Thomas Besnier, Sylvain Arguillère, Emery Pierson, Mohamed Daoudi
Generative Model Point Cloud Generative Modeling Geometric Data Kernel Distance Geodesic Metric Space

April 6, 2023

Parameterized Approximation Schemes for Clustering with General Norm Objectives
Fateme Abbasi, Sandip Banerjee, Jarosław Byrka, Parinya Chalermsook, Ameet Gadekar, Kamyar Khodamoradi, Dániel Marx, Roohani Sharma, Joachim Spoerhase
K$ Mean Parametric Approximation Target Norm K$ Clustering Geodesic Metric Space

February 17, 2023

Minimizing Dynamic Regret on Geodesic Metric Spaces
Zihao Hu, Guanghui Wang, Jacob Abernethy
Sequential Decision Making Problem Dynamic Regret Adaptive Regret Dynamic Regret Minimization Geodesic Metric Space

June 17, 2022

Detecting Adversarial Examples in Batches -- a geometrical approach
Danush Kumar Venkatesh, Peter Steinbach
Adversarial Attack Native Robustness Adversarial Example Adversarial Sample Large Batch Geometric Approach Geodesic Metric Space

June 4, 2022

First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces
Michael I. Jordan, Tianyi Lin, Emmanouil-Vasileios Vlatakis-Gkaragkounis
Min Max Convex Concave First Order Algorithm Strongly Convex Geodesic Metric Space

February 13, 2022

Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm
Peiyuan Zhang, Jingzhao Zhang, Suvrit Sra
Riemannian Gradient Minimax Problem Concave Saddle Point Problem Nonconvex Nonconcave Problem Geodesic Metric Space Minimax Theorem