Concept
Smooth structure
Related lectures (65)
Gradient Descent Methods
Covers gradient descent methods for convex and nonconvex problems, including smooth unconstrained convex minimization, maximum likelihood estimation, and examples like ridge regression and image classification.
Retractions, vector fields and tangent bundles: Retractions and vector fieldsMOOC: Introduction to optimization on smooth manifolds: first order methods
Introduces retractions and vector fields on manifolds, providing examples and discussing smoothness and extension properties.
Dirac Delta Function
Introduces the Dirac delta function and discusses its properties and applications in signal processing and physics.
Optimization Basics: Norms, Convexity, Differentiability
Explores optimization basics such as norms, convexity, and differentiability, along with practical applications and convergence rates.
What is a (sub)manifold
Introduces the concept of a submanifold in a linear space, defining it as a set smoothly embedded in the space.
Riemannian metrics and gradients: Examples and Riemannian submanifoldsMOOC: Introduction to optimization on smooth manifolds: first order methods
Explores Riemannian metrics on manifolds and the concept of Riemannian submanifolds in Euclidean spaces.
Smooth Manifolds: SetupMOOC: Introduction to optimization on smooth manifolds: first order methods
Introduces smooth manifolds, emphasizing the importance of submanifolds of linear spaces.
Newton's method on Riemannian manifolds
Covers Newton's method on Riemannian manifolds, focusing on second-order optimality conditions and quadratic convergence.
Differentiating Vector Fields: How Not to Do ItMOOC: Introduction to optimization on smooth manifolds: first order methods
Discusses the challenges in differentiating vector fields on submanifolds and the importance of choosing the right method.
Analysis IV: Convergence and Approximation in L² Space
Explores convergence and approximation in L² space, emphasizing the limitations of continuous functions and the importance of closed sets.