Optimization on Smooth Manifolds: First-Order Methods
Master the fundamentals of optimization on nonlinear spaces, from manifold basics to Riemannian gradient descent.
Master the fundamentals of optimization on nonlinear spaces, from manifold basics to Riemannian gradient descent.
Dive into the elegant world of optimization on smooth manifolds with this advanced mathematics course. Starting from the foundational question "What is a manifold?", you'll build a robust understanding of submanifolds embedded in real space and their applications in engineering and sciences. Learn to merge smooth geometry with optimization techniques, developing the skills to recognize, analyze, and solve Riemannian optimization problems. By the end of the course, you'll be able to confidently implement and analyze first-order optimization algorithms on manifolds, opening doors to advanced applications in machine learning, computer vision, and signal processing.
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What you'll learn
Recognize and define smooth manifolds in various contexts
Perform calculus operations on manifolds with confidence
Manipulate key concepts from differential and Riemannian geometry
Develop custom geometric tools for specific manifolds of interest
Formulate and analyze Riemannian optimization problems
Implement and analyze first-order Riemannian optimization algorithms
Skills you'll gain
This course includes:
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Graded assignments, exams
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Limited Access access
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There are 6 modules in this course
This course provides a comprehensive introduction to optimization on smooth manifolds, bridging the gap between differential geometry and optimization theory. Students will gain a deep understanding of manifolds, starting from basic definitions and progressing to advanced concepts in Riemannian geometry. The curriculum covers essential topics such as tangent spaces, differentials, retractions, vector fields, and Riemannian gradients. A significant focus is placed on developing the Riemannian gradient descent algorithm, including its analysis and implementation. The course emphasizes both theoretical foundations and practical applications, equipping students with the skills to recognize and formulate Riemannian optimization problems in various fields. The final part introduces students to Manopt, a toolbox for optimization on manifolds, enhancing their ability to prototype and solve real-world problems efficiently. Throughout the course, students will develop geometric intuition and computational skills necessary for advanced work in optimization, machine learning, and related fields.
Introduction
Module 1
Manifolds and tangent spaces
Module 2
Functions, differentials, retractions and vector fields
Module 3
Riemannian manifolds and gradients
Module 4
Riemannian gradient descent
Module 5
Manopt (toolbox for optimization on manifolds)
Module 6
Fee Structure
Instructor
1 Course
Pioneer in Riemannian Optimization and Mathematical Programming
Nicolas Boumal is an Assistant Professor of Mathematics at EPFL, where he specializes in optimization, statistical estimation, and numerical analysis. His groundbreaking work includes developing the award-winning Manopt toolbox for optimization on manifolds. In 2023, he published "An Introduction to Optimization on Smooth Manifolds" with Cambridge University Press, which has become a fundamental text in the field, requiring no prerequisites in geometry or optimization. The book emerged from his graduate course first taught at Princeton University in 2019, covering differential geometry and Riemannian optimization. At EPFL, he teaches Continuous Optimization and Optimization on Manifolds, where he recorded comprehensive lecture series in 2023. His research has advanced the theoretical understanding of Riemannian optimization, particularly in areas such as iteration complexity and geodesic convexity. As an Associate Editor of Mathematical Programming, he continues to shape the field through both research and education
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