An Introduction to Optimization on Smooth Manifolds

An invitation to optimization with Riemannian geometry for applied mathematics, computer science and engineering students and researchers.

Nicolas Boumal (Author)

9781009166157, Cambridge University Press

Paperback / softback, published 16 March 2023

400 pages
25.3 x 17.8 x 2 cm, 0.67 kg

'This new book by Nicolas Boumal focuses on optimization on manifolds, which appears naturally in many areas of data science. It successfully covers all important and required concepts in differential geometry with an intuitive and pedagogical approach which is adapted to readers with no prior exposure. Algorithms and analysis are then presented with the perfect mix of significance and mathematical depth. This is a must-read for all graduate students and researchers in data science.' Francis Bach, INRIA

Optimization on Riemannian manifolds-the result of smooth geometry and optimization merging into one elegant modern framework-spans many areas of science and engineering, including machine learning, computer vision, signal processing, dynamical systems and scientific computing. This text introduces the differential geometry and Riemannian geometry concepts that will help students and researchers in applied mathematics, computer science and engineering gain a firm mathematical grounding to use these tools confidently in their research. Its charts-last approach will prove more intuitive from an optimizer's viewpoint, and all definitions and theorems are motivated to build time-tested optimization algorithms. Starting from first principles, the text goes on to cover current research on topics including worst-case complexity and geodesic convexity. Readers will appreciate the tricks of the trade for conducting research and for numerical implementations sprinkled throughout the book.

Notation
1. Introduction
2. Simple examples
3. Embedded geometry: first order
4. First-order optimization algorithms
5. Embedded geometry: second order
6. Second-order optimization algorithms
7. Embedded submanifolds: examples
8. General manifolds
9. Quotient manifolds
10. Additional tools
11. Geodesic convexity
References
Index.

Subject Areas: Pattern recognition [UYQP], Optimization [PBU], Differential & Riemannian geometry [PBMP]