MAGMA: Multi-level accelerated gradient mirror descent algorithm for large-scale convex composite minimization

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Title: MAGMA: Multi-level accelerated gradient mirror descent algorithm for large-scale convex composite minimization
Authors: Hovhannisyan, V
Parpas, P
Zafeiriou, S
Item Type: Journal Article
Abstract: Composite convex optimization models arise in several applications, and are especially prevalent in inverse problems with a sparsity inducing norm and in general convex optimization with simple constraints. The most widely used algorithms for convex composite models are accelerated first order methods, however they can take a large number of iterations to compute an acceptable solution for large-scale problems. In this paper we propose to speed up first order methods by taking advantage of the structure present in many applications and in image processing in particular. Our method is based on multi-level optimization methods and exploits the fact that many applications that give rise to large scale models can be modelled using varying degrees of fidelity. We use Nesterov’s acceleration techniques together with the multi-level approach to achieve an O(1/ √ ǫ) convergence rate, where ǫ denotes the desired accuracy. The proposed method has a better convergence rate than any other existing multi-level method for convex problems, and in addition has the same rate as accelerated methods, which is known to be optimal for first-order methods. Moreover, as our numerical experiments show, on large-scale face recognition problems our algorithm is several times faster than the state of the art.
Issue Date: 1-Jan-2016
Date of Acceptance: 1-Sep-2016
URI: http://hdl.handle.net/10044/1/40967
DOI: https://dx.doi.org/10.1137/15M104013X
ISSN: 1936-4954
Publisher: Society for Industrial and Applied Mathematics
Start Page: 1829
End Page: 1857
Journal / Book Title: SIAM Journal on Imaging Sciences
Volume: 9
Issue: 4
Copyright Statement: Copyright © by SIAM. Unauthorized reproduction of this article is prohibited
Sponsor/Funder: Commission of the European Communities
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (E
Funder's Grant Number: FP7 - 321698
EP/K040723/1
EP/M028240/1
Keywords: Science & Technology
Technology
Physical Sciences
Computer Science, Artificial Intelligence
Computer Science, Software Engineering
Mathematics, Applied
Imaging Science & Photographic Technology
Computer Science
Mathematics
convex optimization
multilevel optimization
linear inverse problem
optimal gradient methods
face recognition
accelerated proximal gradient method
ROBUST FACE RECOGNITION
LINEAR INVERSE PROBLEMS
SPARSE REPRESENTATION
THRESHOLDING ALGORITHM
NONLINEAR OPTIMIZATION
COORDINATE DESCENT
MULTIGRID METHOD
SHRINKAGE
L(1)-MINIMIZATION
SELECTION
Artificial Intelligence & Image Processing
Publication Status: Published
Appears in Collections:Faculty of Engineering
Computing



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