Course

MATH-512: Optimization on manifolds

Lectures in this course (95)

Explores gradient descent methods for optimizing functions on manifolds, emphasizing small gradient guarantees and global convergence.

Connections: Axiomatic Definition

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

Choosing a Step Size

Explores choosing a step size in optimization on manifolds, including backtracking line-search and the Armijo method.

Riemannian Trust Regions: Global Convergence

Discusses regularity conditions, trust regions, and global convergence in optimization.

Riemannian Trust Regions framework

Introduces the Riemannian Trust Regions (RTR) framework, covering conjugate directions, Newton's method, and model improvement.

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 connections: What they are and why we care

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

Hands on with Manopt

Explores practical optimization using Manopt for manifolds, covering gradient checks, approximation errors, and Hessian computations.

Riemannian connections: Proof sketch

Presents the fundamental theorem of Riemannian geometry and demonstrates the uniqueness of the Riemannian connection.

RTR practical aspects + tCG

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

Grassmann manifold and Retractions

Covers the Grassmann manifold and retractions on submanifolds.

Riemannian Hessians: Connections and Symmetry

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

Riemannian Hessians: Definition and Example

Covers the definition and computation of Riemannian Hessians on manifolds.

General Manifolds and Topology

Covers manifolds, topology, smooth maps, and tangent vectors in detail.

General Manifolds: A First Look

Introduces d-dimensional charts on manifolds and the concept of an atlas.

Differentiating Vector Fields: Definition

Introduces differentiating vector fields along curves on manifolds with connections and the unique operator satisfying specific properties.

Differentiating vector fields along curves: Examples and finite differences

Covers the differentiation of vector fields along curves with examples and finite differences.

Geodesically Convex Optimization

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

Acceleration and geodesics

Explains acceleration along curves and geodesics on manifolds, generalizing straight lines to spheres.

Riemannian distance, geodesically convex sets

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