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A Limited-Memory Riemannian Symmetric Rank-One Trust-Region Method with a Restart Strategy

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Abstract

Limited-memory versions of quasi-Newton methods are efficient approaches to solving 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.

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Data Availability

The datasets generated during and analysed during the current study are available from the corresponding author on reasonable request.

Notes

  1. A preliminary version of the paper can be found in [21]. This paper is different from [21] in many places. In [21], i) it is not shown that all the Hessian approximations are bounded from above. Therefore, the global convergence is guaranteed by the heuristic of truncating all Hessian approximations if their norms are too large, ii) the vector transport may not be isometric, iii) the complexity analysis and implementation are not derived and described, iv) the LRTRSR1 with restarting using intrinsic representation is not given, v) only preliminary experiments are performed.

  2. In [4], a variant of the subproblem is used, i.e., the norm for defining the radius varies in every iteration.

  3. Conventionally, a vector transport is defined to be a smooth mapping, see e.g., [2]. Here, we do not require such a property by default and the smoothness is imposed only when needed.

  4. The pseudo-inverse is independent of the Riemannian metric. It is the conventional pseudo-inverse with respect to the Euclidean metric in \({\mathbb {R}}^n\).

  5. Note that in this case, \(y_k^T s_k\) is always positive since \(y_k^T s_k = s_k^T A s_k > 0\).

  6. In this case, \(y_k = {\bar{G}}_k s_k\), where \({\bar{G}}_k = \int _0^1 \nabla ^2 f(x_{k} + t(x_{k+1} - x_k) ) dt\) denotes the average Hessian. If \(x_k\) and \(x_{k+1}\) are sufficiently close to \(x_*\), then the average Hessian can be sufficiently close to \(\nabla f(x_*)\) such that \({\bar{G}}_k\) is symmetric positive definite. It follows that \(y_k^T s_k = s_k^T {\bar{G}}_k s_k >0\).

  7. Intuitively, it is natural to find a nonnegative root of the function \(\Vert \varphi (\sigma )\Vert _2 - \varDelta _k\). However, this function may cause numerical difficulties. We refer to [31, pp. 85 and 86] for more details.

  8. Theoretically, \(\sigma _*\) needs to be the exact root. However, such \(\sigma _*\) is impractical numerically due to numerical error and the accuracy of the Newton method. We choose this stopping criterion since it usually reaches high accuracy and is robust. In our numerical experiments, the number of iterations of the Newton’s method for finding \(\sigma _*\) is usually round 3 and the final magnitude of \(\phi \) is usually around \(10^{-16}\) or lower. Occasionally, the number of iterations reaches the maximum \(2 \ell + 10\). In these cases, the final magnitudes of \(\phi \) are usually around \(10^{-10}\).

  9. Version of August 14 2012 is available at: http://lmafit.blogs.rice.edu

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Funding

Wen Huang was partially supported by the Fundamental Research Funds for the Central Universities (NO. 20720190060) and National Natural Science Foundation of China (NO. 12001455).

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  1. School of Mathematical Sciences, Xiamen University, Xiamen, China

    Wen Huang

  2. Department of Mathematics, Florida State University, Tallahassee, FL, USA

    Kyle A. Gallivan

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A preliminary version of this paper can be found in [21]

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Huang, W., Gallivan, K.A. A Limited-Memory Riemannian Symmetric Rank-One Trust-Region Method with a Restart Strategy. J Sci Comput 93, 1 (2022). https://doi.org/10.1007/s10915-022-01962-0

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