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Non-convex Rank/Sparsity Regularization and Local Minima

Olsson, Carl LU ; Carlsson, Marcus LU ; Andersson, Fredrik LU and Larsson, Viktor LU (2017) 16th IEEE International Conference on Computer Vision, ICCV 2017 In Proceedings - 2017 IEEE International Conference on Computer Vision, ICCV 2017 2017-October. p.332-340
Abstract

This paper considers the problem of recovering either a low rank matrix or a sparse vector from observations of linear combinations of the vector or matrix elements. Recent methods replace the non-convex regularization with ℓ1 or nuclear norm relaxations. It is well known that this approach recovers near optimal solutions if a so called restricted isometry property (RIP) holds. On the other hand it also has a shrinking bias which can degrade the solution. In this paper we study an alternative non-convex regularization term that does not suffer from this bias. Our main theoretical results show that if a RIP holds then the stationary points are often well separated, in the sense that their differences must be of high cardinality/rank.... (More)

This paper considers the problem of recovering either a low rank matrix or a sparse vector from observations of linear combinations of the vector or matrix elements. Recent methods replace the non-convex regularization with ℓ1 or nuclear norm relaxations. It is well known that this approach recovers near optimal solutions if a so called restricted isometry property (RIP) holds. On the other hand it also has a shrinking bias which can degrade the solution. In this paper we study an alternative non-convex regularization term that does not suffer from this bias. Our main theoretical results show that if a RIP holds then the stationary points are often well separated, in the sense that their differences must be of high cardinality/rank. Thus, with a suitable initial solution the approach is unlikely to fall into a bad local minimum. Our numerical tests show that the approach is likely to converge to a better solution than standard ℓ1/nuclear-norm relaxation even when starting from trivial initializations. In many cases our results can also be used to verify global optimality of our method.

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author
organization
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type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
Proceedings - 2017 IEEE International Conference on Computer Vision, ICCV 2017
volume
2017-October
pages
9 pages
publisher
Institute of Electrical and Electronics Engineers Inc.
conference name
16th IEEE International Conference on Computer Vision, ICCV 2017
external identifiers
  • scopus:85041894852
ISBN
9781538610329
DOI
10.1109/ICCV.2017.44
language
English
LU publication?
yes
id
e010ffd0-9073-4a0e-bd12-799fa82fe3d1
date added to LUP
2018-02-22 09:04:02
date last changed
2018-05-29 11:41:01
@inproceedings{e010ffd0-9073-4a0e-bd12-799fa82fe3d1,
  abstract     = {<p>This paper considers the problem of recovering either a low rank matrix or a sparse vector from observations of linear combinations of the vector or matrix elements. Recent methods replace the non-convex regularization with ℓ1 or nuclear norm relaxations. It is well known that this approach recovers near optimal solutions if a so called restricted isometry property (RIP) holds. On the other hand it also has a shrinking bias which can degrade the solution. In this paper we study an alternative non-convex regularization term that does not suffer from this bias. Our main theoretical results show that if a RIP holds then the stationary points are often well separated, in the sense that their differences must be of high cardinality/rank. Thus, with a suitable initial solution the approach is unlikely to fall into a bad local minimum. Our numerical tests show that the approach is likely to converge to a better solution than standard ℓ1/nuclear-norm relaxation even when starting from trivial initializations. In many cases our results can also be used to verify global optimality of our method.</p>},
  author       = {Olsson, Carl and Carlsson, Marcus and Andersson, Fredrik and Larsson, Viktor},
  booktitle    = {Proceedings - 2017 IEEE International Conference on Computer Vision, ICCV 2017},
  isbn         = {9781538610329},
  language     = {eng},
  month        = {12},
  pages        = {332--340},
  publisher    = {Institute of Electrical and Electronics Engineers Inc.},
  title        = {Non-convex Rank/Sparsity Regularization and Local Minima},
  url          = {http://dx.doi.org/10.1109/ICCV.2017.44},
  volume       = {2017-October},
  year         = {2017},
}