Riemannian Hessians: Definition and Example

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Riemannian metrics and gradients: Why and definition of Riemannian manifolds

Covers Riemannian metrics, gradients, vector fields, and inner products on manifolds.

Taylor expansions: second order

Explores Taylor expansions and retractions on Riemannian manifolds, emphasizing second-order approximations and covariant derivatives.

Riemannian metrics and gradients: Riemannian gradients

Explains Riemannian submanifolds, metrics, and gradients computation on manifolds.

Connections: motivation and definition

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

Newton's method on Riemannian manifolds

Covers Newton's method on Riemannian manifolds, focusing on second-order optimality conditions and quadratic convergence.

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.

Bird's-eye view and aims

Covers optimization on manifolds, smoothness, tools needed for optimization, and advanced algorithms using Hessians and Riemannian connections.

Linear convergence with Polyak-Łojasiewicz: Mechanical proof

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

RTR practical aspects + tCG

Explores practical aspects of Riemannian trust-region optimization and introduces the truncated conjugate gradient method.

Riemannian distance, geodesically convex sets

Covers the structure of Riemannian manifolds, geodesic convexity, and the Riemannian distance function.