Optimal convergence rates for generalized alternating projections
Fält, Mattias LU and Giselsson, Pontus LU
(2017)
56th IEEE Annual Conference on Decision and Control, CDC 2017
p.2268-2274
- Abstract
- Generalized alternating projections is an algorithm that alternates relaxed projections onto a finite number of sets to find a point in their intersection. We consider the special case of two linear subspaces, for which the algorithm reduces to matrix multiplications. For convergent powers of the matrix, the asymptotic rate is linear and decided by the magnitude of the subdominant eigenvalue. In this paper, we show how to select the three algorithm parameters to optimize this magnitude, and hence the asymptotic convergence rate. The obtained rate depends on the Friedrichs angle between the subspaces and is considerably better than known rates for other methods such as alternating projections and DouglasRachford splitting. We also present... (More)
- Generalized alternating projections is an algorithm that alternates relaxed projections onto a finite number of sets to find a point in their intersection. We consider the special case of two linear subspaces, for which the algorithm reduces to matrix multiplications. For convergent powers of the matrix, the asymptotic rate is linear and decided by the magnitude of the subdominant eigenvalue. In this paper, we show how to select the three algorithm parameters to optimize this magnitude, and hence the asymptotic convergence rate. The obtained rate depends on the Friedrichs angle between the subspaces and is considerably better than known rates for other methods such as alternating projections and DouglasRachford splitting. We also present an adaptive scheme that, online, estimates the Friedrichs angle and updates the algorithm parameters based on this estimate. A numerical example is provided that supports our theoretical claims and shows very good performance for the adaptive method. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/edbb4c52-621f-4ace-a8b9-61e0d5a0660f
- author
- Fält, Mattias
LU
and Giselsson, Pontus
LU
- organization
- publishing date
- 2017-12-12
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- keywords
- OPTIMIZATION, First order optimization algorithms
- host publication
- Proceedings of the IEEE Conference on Decision and Control, 2017
- pages
- 7 pages
- publisher
- IEEE - Institute of Electrical and Electronics Engineers Inc.
- conference name
- 56th IEEE Annual Conference on Decision and Control, CDC 2017
- conference location
- Melbourne, Australia
- conference dates
- 2017-12-12 - 2017-12-15
- external identifiers
-
- scopus:85046282732
- ISBN
- 978-1-5090-2873-3
- 978-1-5090-2874-0
- DOI
- 10.1109/CDC.2017.8263980
- language
- English
- LU publication?
- yes
- id
- edbb4c52-621f-4ace-a8b9-61e0d5a0660f
- alternative location
- https://arxiv.org/abs/1703.10547
- date added to LUP
- 2018-02-01 09:31:54
- date last changed
- 2025-01-07 04:19:40
@inproceedings{edbb4c52-621f-4ace-a8b9-61e0d5a0660f,
abstract = {{Generalized alternating projections is an algorithm that alternates relaxed projections onto a finite number of sets to find a point in their intersection. We consider the special case of two linear subspaces, for which the algorithm reduces to matrix multiplications. For convergent powers of the matrix, the asymptotic rate is linear and decided by the magnitude of the subdominant eigenvalue. In this paper, we show how to select the three algorithm parameters to optimize this magnitude, and hence the asymptotic convergence rate. The obtained rate depends on the Friedrichs angle between the subspaces and is considerably better than known rates for other methods such as alternating projections and DouglasRachford splitting. We also present an adaptive scheme that, online, estimates the Friedrichs angle and updates the algorithm parameters based on this estimate. A numerical example is provided that supports our theoretical claims and shows very good performance for the adaptive method.}},
author = {{Fält, Mattias and Giselsson, Pontus}},
booktitle = {{Proceedings of the IEEE Conference on Decision and Control, 2017}},
isbn = {{978-1-5090-2873-3}},
keywords = {{OPTIMIZATION; First order optimization algorithms}},
language = {{eng}},
month = {{12}},
pages = {{2268--2274}},
publisher = {{IEEE - Institute of Electrical and Electronics Engineers Inc.}},
title = {{Optimal convergence rates for generalized alternating projections}},
url = {{http://dx.doi.org/10.1109/CDC.2017.8263980}},
doi = {{10.1109/CDC.2017.8263980}},
year = {{2017}},
}