Riemannian Optimization and Its Applications

This brief describes the basics of Riemannian optimization—optimization on Riemannian manifolds—introduces algorithms for Riemannian optimization problems, discusses the theoretical properties of these algorithms, and suggests possible applications of Riemannian optimization to problems in other fields.

To provide the reader with a smooth introduction to Riemannian optimization, brief reviews of mathematical optimization in Euclidean spaces and Riemannian geometry are included. Riemannian optimization is then introduced by merging these concepts. In particular, the Euclidean and Riemannian conjugate gradient methods are discussed in detail. A brief review of recent developments in Riemannian optimization is also provided.
 
Riemannian optimization methods are applicable to many problems in various fields. This brief discusses some important applications including the eigenvalue and singular value decompositions in numericallinear algebra, optimal model reduction in control engineering, and canonical correlation analysis in statistics.

“The author successfully presents all of this varied material using a consistent and modern notation. … The book meticulously provides references with a comprehensive list at the end. It includes information about software libraries that implement Riemannian optimization in MATLAB, Python, R, C++, and Julia. Both the proofs and calculations in the examples are given with sufficient detail using a consistent notation.” (Anders Linnér, Mathematical Reviews, October, 2022)

“The book is a very nice introductory reference for students, engineers, and practitioners to get started in the field of Riemannian optimization. … A highlight of the book is that it reviews the most important work in the field and also mentions current research topics. Thus, I also highly recommended it to researchers getting a broad overview of what is currently studied in the field, without being too detailed or theoretical.” (Lena Sembach, SIAM Review, Vol. 64 (2), June, 2022)

Doctor Hiroyuki Sato received his bachelor of engineering degree from the Faculty of Engineering, Kyoto University, Japan, in 2009 and his master and doctor of informatics degrees from the Graduate School of Informatics, Kyoto University, in 2011 and 2013, respectively. Following graduation, from 2013 to 2014, he was a research fellow of the Japan Society for the Promotion of Science. Subsequently, from 2014 to 2017, he worked as an assistant professor in the Faculty of Engineering, Tokyo University of Science, Japan. Further, from 2017 to 2018, Doctor Sato was employed as an assistant professor in the Hakubi Center for Advanced Research, Kyoto University. Currently, he is an associate professor in the Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University.

Doctor Sato has been studying Riemannian optimization, that is, geometric optimization on Riemannian manifolds. His research interests include the theory of Riemannian optimization algorithms, such as the Riemannian conjugate gradient methods, and their applications to problems in other fields. His works contribute to developing the Riemannian optimization theory and proposing Riemannian optimization algorithms for problems arising in applications. Because Riemannian optimization is an interdisciplinary research field with diverse applications, Doctor Sato routinely collaborates with researchers across various fields, including numerical linear algebra, control engineering, and statistics.