Riemannian Gradient Descent: Convergence Theorem and Line Search Method

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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.

Geodesically Convex Optimization

Covers geodesically convex optimization on Riemannian manifolds, exploring convexity properties and minimization relationships.

Optimization on Manifolds

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

Computing the Newton Step: Matrix-Based Approaches

Explores matrix-based approaches for computing the Newton step on a Riemannian manifold.

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.

Riemannian distance, geodesically convex sets

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

Riemannian Hessians: Definition and Example

Covers the definition and computation of Riemannian Hessians on manifolds.

Taylor expansions: second order

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

Riemannian Hessians: Connections and Symmetry

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

Newton's method on Riemannian manifolds

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