A limited-memory Riemannian symmetric rank-one trust-region method with a restart strategy

Authors

Wen Huang*, Kyle A. Gallivan

Abstract

Limited-memory versions of quasi-Newton methods are efficient methods for large-scale optimization problems in a Euclidean space. In particular, a quasi-Newton symmetric rank-one update used in a trust-region setting has proven to be an effective method. In this paper, a limited-memory Riemannian symmetric rank-one trust-region method with a restart strategy is proposed by combining the intrinsic representation of tangent vectors with a recently proposed efficient solver for the Euclidean limited-memory symmetric rank-one trust-region subproblem. The global convergence is established under the assumptions that the vector transport is isometric and the cost function is Lipschitz continuously differentiable. The computational and spatial complexity are analyzed, and detailed implementations are described. The performance of the proposed method is compared to limited-memory Riemannian BFGS method and other state-of-the-art methods using problems from matrix completion, independent component analysis, phase retrieval, and blind deconvolution. The proposed method is novel for problems on Riemannian and Euclidean spaces.

Key words

Riemannian Optimization; limited-memory quasi-Newton; trust-region; Symmetric rank-one; Matrix completion; Joint diagonalization; Phase retrieval; Blind deconvolution;

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