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MOOC# Introduction to optimization on smooth manifolds: first order methods

Description

Optimization on manifolds is the result of smooth geometry and optimization merging into one elegant modern framework.

We start the course at "What is a manifold?", and give the students a firm understanding of submanifolds embedded in real space. This covers numerous applications in engineering and the sciences.

All definitions and theorems are motivated to build time-tested optimization algorithms. The math is precise, to promote understanding and enable computation.

We build our way up to Riemannian gradient descent: the all-important first-order optimization algorithm on manifolds. This includes analysis and implementation.

The lectures follow (and complement) the textbook "An introduction to optimization on smooth manifolds" written by the instructor, also available on his webpage.

From there, students can explore more with numerical tools (such as the toolbox Manopt, which is the subject of the last week of the course). They will also be in a good position to tackle more adv

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Lectures in this MOOC (66)

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Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or function).

Pseudo-Riemannian manifold

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Differentiable manifold

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