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Riemannian Optimization and Its Applications

Part of the SpringerBriefs in Electrical and Computer Engineering Book Series

DOI: 10.1007/978-3-030-62391-3

ISBN: 978-3-030-62389-0

Publisher: Springer International Publishing - Springer Journals

Feb 18, 2021

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Introduction

Sato, Hiroyuki

Pages: 1 –11

[In this chapter, we focus on unconstrained and constrained minimization problems involving a quadratic function as simple examples of optimization problems. Then, we observe that a problem with the constraint that the sum of the squares of its decision variables should be one can be regarded as...
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Preliminaries and Overview of Euclidean Optimization

Sato, Hiroyuki

Pages: 13 –36

[In this chapter, we briefly review some mathematical concepts as preliminaries for our discussion of Riemannian optimization; these concepts range from linear algebra to general topology and include Euclidean optimization. Although readers are expected to have basic knowledge of linear algebra...
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Unconstrained Optimization on Riemannian Manifolds

Sato, Hiroyuki

Pages: 37 –64

[In this chapter, we introduce the concept of a Riemannian manifold, based on which Riemannian optimization is developed. We also introduce the concept of a retraction, which is important when searching for the next point in an optimization procedure. Furthermore, using a retraction, we discuss...
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Conjugate Gradient Methods on Riemannian Manifolds

Sato, Hiroyuki

Pages: 65 –88

[In this chapter, we discuss the conjugate gradient (CG) methods on Riemannian manifolds, which we also call Riemannian CG methods. They can be considered to be a modified version of the Riemannian steepest descent method. In particular, we analyze the Fletcher–Reeves-type and Dai–Yuan-type...
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Applications of Riemannian Optimization

Sato, Hiroyuki

Pages: 89 –106

[In this chapter, we introduce some practical examples of Riemannian optimization. The introduced problems are themselves important; furthermore, they can provide readers with insights into other Riemannian optimization problems and help them in solving such problems that they may encounter.]
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Recent Developments in Riemannian Optimization

Sato, Hiroyuki

Pages: 107 –115

[In this chapter, we review the recent developments in Riemannian optimization, such as stochastic and constrained optimization. A few other topics, including second-order and nonsmooth optimization, are also briefly reviewed. Interested readers may refer to the references introduced in the...

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