The content delves into the fallacy of employing a single tangent space for operations on Riemannian manifolds in robot learning. It discusses the limitations of this approach, provides theoretical insights, and presents experimental evidence showcasing its shortcomings. The paper emphasizes best practices for utilizing Riemannian methods effectively in robot learning applications.
The analysis covers various aspects such as density estimation on spheres and SPD manifolds, learning dynamical systems with normalizing flows, and evaluating performance metrics like DTWD and success rates. It underscores the significance of designing coordinate-invariant algorithms, leveraging a bundle of tangent spaces, and formulating learning models based on Riemannian theory.
Key points include discussions on geometric constraints in robotics data, misconceptions surrounding single tangent space approaches, mathematical explanations of fallacies, experiments illustrating performance differences among Euclidean GMMs, Tangent GMMs, and Riemannian GMMs. The content concludes by advocating for mathematically sound techniques to unlock the full potential of Riemannian manifolds in robot learning.
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