Riemannian Metric

Riemannian metrics define distance and curvature on manifolds, extending Euclidean geometry to non-flat spaces crucial for modeling complex data like images, covariance matrices, and robot configurations. Current research focuses on developing computationally efficient metrics for various manifolds (e.g., Stiefel, Cholesky, SPD), often incorporating neural networks for adaptive metric learning and improved optimization in applications like deep learning and motion planning. This work is significant because it enables the development of more accurate and efficient algorithms for diverse applications, ranging from image processing and brain connectome analysis to robotics and machine learning.