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.

‘With its inviting embedded-first progression and its many examples and exercises, this book constitutes an excellent companion to the literature on Riemannian optimization - from the early developments in the late 20th century to topics that have gained prominence since the 2008 book ‘Optimization Algorithms on Matrix Manifolds’, and related software, such as Manopt/Pymanopt/Manopt.jl.’

P.-A. Absil - University of Louvain

‘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