A.Y. 2020/2021
Differential Geometry in Applications
manifold learning, optimization and information geometry
Mathematical Models and Methods in Engineering
Lecturers
Samuele Mongodi
CONTENTS
- Submanifolds of $\mathbb{R}^n$: geometry and optimization problems (gradient descent on submanifolds).
- Riemannian Geometry: metrics and connections (Fisher metric), vectorfields-differential forms duality (gradient descent v2), geodesics and distances (dimensionality reduction algorithms: ISOMAP, LLE. …)
- Curvatures: geometric meaning, Laplace-Beltrami operator and its eigenvalues (Eigenmap and diffusion algorithms for dimensionality reduction).
- Discretization: sampling on manifolds (graphs, triangulations, meshes…), brief overview of Hamiltonian dynamics and Monte Carlo gradient descent.
