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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.

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organization
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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
2019-03-19 03:51:26
@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},
  keyword      = {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},
  volume       = {63},
  year         = {2018},
}