Low-Rank Optimization with Convex Constraints
Grussler, Christian LU ; Rantzer, Anders LU
and Giselsson, Pontus
LU
(2018)
In IEEE Transactions on Automatic Control
63(11).
p.4000-4007
- Abstract
The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design and low-complexity modelling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints, while minimizing the distance to the matrix in the squared Frobenius norm. In many situations, this non-convex problem is convexified by nuclear norm regularization. However, we will see that the approximations obtained by this method may be far from optimal. Here, we propose an alternative convex relaxation that uses the convex envelope of the squared Frobenius norm and the rank constraint. With this approach, easily... (More)
The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design and low-complexity modelling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints, while minimizing the distance to the matrix in the squared Frobenius norm. In many situations, this non-convex problem is convexified by nuclear norm regularization. However, we will see that the approximations obtained by this method may be far from optimal. Here, we propose an alternative convex relaxation that uses the convex envelope of the squared Frobenius norm and the rank constraint. With this approach, easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original non-convex problem coincide. An SDP representation of the convex envelope is derived, which allows us to treat several known problems. Our example on optimal low-rank Hankel approximation/model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear norm regularization as well as balanced truncation.
(Less)
- author
- Grussler, Christian
LU
; Rantzer, Anders
LU
and Giselsson, Pontus
LU
- organization
- publishing date
- 2018-03-07
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Analytical models, Compressed Sensing, Convex functions, Data models, k-support norm, Low-rank Approximation, Mathematical model, Model Reduction, Optimization, Radio frequency, Reduced order systems, System Identification
- in
- IEEE Transactions on Automatic Control
- volume
- 63
- issue
- 11
- pages
- 4000 - 4007
- publisher
- IEEE - Institute of Electrical and Electronics Engineers Inc.
- external identifiers
-
- scopus:85043360737
- ISSN
- 0018-9286
- DOI
- 10.1109/TAC.2018.2813009
- language
- English
- LU publication?
- yes
- id
- e6dc0d36-59c9-4e4f-aea4-55edd1fc5fbf
- date added to LUP
- 2018-03-20 09:18:59
- date last changed
- 2023-11-17 15:51:02
@article{e6dc0d36-59c9-4e4f-aea4-55edd1fc5fbf,
abstract = {{<p>The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design and low-complexity modelling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints, while minimizing the distance to the matrix in the squared Frobenius norm. In many situations, this non-convex problem is convexified by nuclear norm regularization. However, we will see that the approximations obtained by this method may be far from optimal. Here, we propose an alternative convex relaxation that uses the convex envelope of the squared Frobenius norm and the rank constraint. With this approach, easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original non-convex problem coincide. An SDP representation of the convex envelope is derived, which allows us to treat several known problems. Our example on optimal low-rank Hankel approximation/model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear norm regularization as well as balanced truncation.</p>}},
author = {{Grussler, Christian and Rantzer, Anders and Giselsson, Pontus}},
issn = {{0018-9286}},
keywords = {{Analytical models; Compressed Sensing; Convex functions; Data models; k-support norm; Low-rank Approximation; Mathematical model; Model Reduction; Optimization; Radio frequency; Reduced order systems; System Identification}},
language = {{eng}},
month = {{03}},
number = {{11}},
pages = {{4000--4007}},
publisher = {{IEEE - Institute of Electrical and Electronics Engineers Inc.}},
series = {{IEEE Transactions on Automatic Control}},
title = {{Low-Rank Optimization with Convex Constraints}},
url = {{http://dx.doi.org/10.1109/TAC.2018.2813009}},
doi = {{10.1109/TAC.2018.2813009}},
volume = {{63}},
year = {{2018}},
}