Geodesic convexity: why, and what we need

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Explores matrix-based and matrix-free approaches for computing the Newton step in optimization on manifolds.

Explores connections on manifolds, emphasizing the axiomatic definition and properties of derivatives in differentiating vector fields.

Explores gradient descent with memory, momentum methods, conjugate gradients, and nonlinear CG on manifolds.

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

Explores the importance of comparing tangent vectors at different points using algorithms and finite differences.

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

Explores necessary and sufficient optimality conditions for local minima on manifolds, focusing on second-order critical points.

Covers the concept of lines spaces and equivalent matrices in linear algebra.

Explores smooth maps, differentials, composition properties, linearity, and extensions on manifolds.

Introduces Manopt, a toolbox for optimization on manifolds, covering gradient and Hessian checks, solver calls, and manual caching.