Cayley-transform-based gradient and conjugate gradient algorithms on Grassmann manifolds

Zhu, Xiaojing; Sato, Hiroyuki

doi:10.1007/s10444-021-09880-9

Cited by 7 publications

(2 citation statements)

References 34 publications

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“…By making use of these tools, the constrained problem (1.14) can be treated as an unconstrained problem on manifolds. From this viewpoint, some classical unconstrained optimization methods in Euclidean setting have been extended to the setting of Riemannian optimization, such as the gradient method, trust region methods, CG methods, and quasi-Newton methods; see [22][23][24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

A class of spectral conjugate gradient methods for Riemannian optimization

et al. 2023

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Spectral conjugate gradient (SCG) methods are combinations of spectral gradient method and conjugate gradient (CG) methods, which have been well studied in Euclidean space. In this paper, we aim to extend this class of methods to solve optimization problems on Riemannian manifolds. Firstly, we present a Riemannian version of the spectral parameter, which guarantees that the search direction always satisfies the sufficient descent property without the help of any line search strategy. Secondly, we introduce a generic algorithmic framework for the Riemannian SCG methods, in which the selection of the CG parameter is very flexible. Under the Riemannian Wolfe conditions, the global convergence of the proposed algorithmic framework is established whenever the absolute value of the CG parameter is no more than the Riemannian Fletcher-Reeves CG parameter. Finally, some preliminary numerical results are reported and compared with several classical Riemannian CG methods, which show that our new methods are efficient.

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Section: Introductionmentioning

confidence: 99%

A class of spectral conjugate gradient methods for Riemannian optimization

et al. 2023

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“…Another example is the manifold SPD(n), which comprises all n \times n symmetric positive definite matrices [15,42]. Furthermore, the class of unconstrained Riemannian optimization problems also covers problems that are not defined in Euclidean space, e.g., optimization problems on the Grassmann manifold Grass(p, n) := \{ W \subset \BbbR n | W is a p-dimensional subspace of \BbbR n \} for p \leq n, whose decision variable W is a subspace of \BbbR n , not a vector or matrix [22,64].…”

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confidence: 99%

Riemannian Conjugate Gradient Methods: General Framework and Specific Algorithms with Convergence Analyses

Sato¹

2022

SIAM J. Optim.

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Conjugate gradient methods are important first-order optimization algorithms both in Euclidean spaces and on Riemannian manifolds. However, while various types of conjugate gradient methods have been studied in Euclidean spaces, there are relatively fewer studies for those on Riemannian manifolds (i.e., Riemannian conjugate gradient methods). This paper proposes a novel general framework that unifies existing Riemannian conjugate gradient methods such as the ones that utilize a vector transport or inverse retraction. The proposed framework also develops other methods that have not been covered in previous studies. Furthermore, conditions for the convergence of a class of algorithms in the proposed framework are clarified. Moreover, the global convergence properties of several specific types of algorithms are extensively analyzed. The analysis provides the theoretical results for some algorithms in a more general setting than the existing studies and new developments for other algorithms. Numerical experiments are performed to confirm the validity of the theoretical results. The experimental results are used to compare the performances of several specific algorithms in the proposed framework.

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Practical gradient and conjugate gradient methods on flag manifolds

Zhu,

Shen

2024

Comput Optim Appl

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Cayley-transform-based gradient and conjugate gradient algorithms on Grassmann manifolds

Cited by 7 publications

References 34 publications

A class of spectral conjugate gradient methods for Riemannian optimization

A class of spectral conjugate gradient methods for Riemannian optimization

Riemannian Conjugate Gradient Methods: General Framework and Specific Algorithms with Convergence Analyses

Practical gradient and conjugate gradient methods on flag manifolds

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