Lectures in course MATH-512: Optimization on manifolds

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

Introduces optimization on manifolds, covering classical and modern techniques in the field.

Explores tangent spaces as free movement directions on submanifolds, offering a geometrically satisfying linearization notion.

Delves into optimization on manifolds, showcasing simple applications like finding largest eigenvalues and singular values.

Explores how submanifolds can be deformed into linear patches using diffeomorphisms and the inverse function theorem.

Explores the applications of Gaussian Mixture Models in recommender systems and real-world scenarios like the Roomba robot.

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

Introduces the concept of a submanifold in a linear space, defining it as a set smoothly embedded in the space.

Introduces smooth manifolds, emphasizing the importance of submanifolds of linear spaces.

Explores fixed-rank matrices as an embedded submanifold, focusing on constructing local defining functions and efficiently computing tangent vectors.

Explores smooth manifolds through defining functions and submanifolds, highlighting the importance of smoothness.

Covers embedded submanifolds, Stiefel manifold, tangent spaces, and differential ranks.

Explores smooth manifolds through diffeomorphisms and embedded submanifolds in a linear space.

Covers smooth maps, vector fields, and retractions on manifolds, emphasizing the importance of smoothly varying curves.

Covers smooth maps on manifolds, defining functions, tangent spaces, and differentials.

Introduces tangent spaces to submanifolds and their properties in optimization on manifolds.

Explores smooth maps between submanifolds, discussing smoothness criteria and composition preservation.

Discusses gradients on Riemannian submanifolds and the construction of local frames.

Covers the definition of retraction, open submanifolds, local defining functions, tangent spaces, and Riemannian metrics.

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