Optimality Conditions: First Order

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Hands on with Manopt: Optimization on Manifolds

Introduces Manopt, a toolbox for optimization on smooth manifolds with a Riemannian structure, covering cost functions, different types of manifolds, and optimization principles.

Optimization on Manifolds

Covers optimization on manifolds, focusing on smooth manifolds and functions, and the process of gradient descent.

Connections: motivation and definition

Explores the definition of connections for smooth vector fields on manifolds.

Algebraic Topology and Differential Geometry

Explores algebraic topology and differential geometry in understanding robot dynamics and global behavior.

Geodesics in Wasserstein Space

Explores geodesics in the Wasserstein space, emphasizing constant speed geodesics and their properties.

Riemannian connections: What they are and why we care

Covers Riemannian connections, emphasizing their properties and significance in geometry.

Riemannian Hessians: Connections and Symmetry

Covers connections on manifolds, symmetric connections, Lie brackets, and compatibility with the metric in Riemannian geometry.

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Linear convergence with Polyak-Łojasiewicz: Mechanical proof

Explores linear convergence with the Polyak-Łojasiewicz condition on a Riemannian manifold.

Retractions vector fields and tangent bundles: Tangent bundles

Covers retractions, tangent bundles, and embedded submanifolds on manifolds with proofs and examples.