Abstract
The circumcentered Douglas–Rachford method (C–DRM), introduced by Behling, Bello Cruz and Santos, iterates by taking the circumcenter of associated successive reflections. It is an acceleration of the well-known Douglas-Rachford method (DRM) for finding the best approximation onto the intersection of finitely many affine subspaces. Inspired by the C–DRM, we introduced the more flexible circumcentered reflection method (CRM) and circumcentered isometry method (CIM). The CIM essentially chooses the closest point to the solution among all of the points in an associated affine hull as its iterate and is a generalization of the CRM. The circumcentered–reflection method introduced by Behling, Bello Cruz and Santos to generalize the C–DRM is a special class of our CRM. We consider the CIM induced by a set of finitely many isometries for finding the best approximation onto the intersection of fixed point sets of the isometries which turns out to be an intersection of finitely many affine subspaces. We extend our previous linear convergence results on CRMs in finite-dimensional spaces from reflections to isometries. In order to better accelerate the symmetric method of alternating projections (MAP), the accelerated symmetric MAP first applies another operator to the initial point. (Similarly, to accelerate the DRM, the C–DRM first applies another operator to the initial point as well.) Motivated by these facts, we show results on the linear convergence of CIMs in Hilbert spaces with first applying another operator to the initial point. In particular, under some restrictions, our results imply that some CRMs attain the known linear convergence rate of the accelerated symmetric MAP in Hilbert spaces. We also exhibit a class of CRMs converging to the best approximation in Hilbert spaces with a convergence rate no worse than the sharp convergence rate of MAP. The fact that some CRMs attain the linear convergence rate of MAP or accelerated symmetric MAP is entirely new.
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Notes
Recall that we use the empty product convention that \({\prod }^{0}_{j=1}\mathrm {R}_{U_{i_{j}}} =\text {Id}\), so the 0-combination of the set I is the Id.
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Acknowledgments
The authors thank the anonymous referees and the editors for their valuable comments and suggestions. HHB and XW were partially supported by NSERC Discovery Grants.
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Bauschke, H., Ouyang, H. & Wang, X. On the linear convergence of circumcentered isometry methods. Numer Algor 87, 263–297 (2021). https://doi.org/10.1007/s11075-020-00966-x
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DOI: https://doi.org/10.1007/s11075-020-00966-x
Keywords
- Isometry
- Projector
- Reflector
- Friedrichs angle
- Best approximation problem
- Linear convergence
- Circumcentered isometry method
- Circumcentered reflection method
- Method of alternating projections
- Accelerated symmetric method of alternating projections