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A Non-convex Relaxation for Fixed-Rank Approximation

Olsson, Carl LU ; Carlsson, Marcus LU and Bylow, Erik LU (2018) 16th IEEE International Conference on Computer Vision Workshops, ICCVW 2017 In Proceedings - 2017 IEEE International Conference on Computer Vision Workshops, ICCVW 2017 2018-January. p.1809-1817
Abstract

This paper considers the problem of finding a low rank matrix from observations of linear combinations of its elements. It is well known that if the problem fulfills a restricted isometry property (RIP), convex relaxations using the nuclear norm typically work well and come with theoretical performance guarantees. On the other hand these formulations suffer from a shrinking bias that can severely degrade the solution in the presence of noise. In this theoretical paper we study an alternative non-convex relaxation that in contrast to the nuclear norm does not penalize the leading singular values and thereby avoids this bias. We show that despite its non-convexity the proposed formulation will in many cases have a single stationary point... (More)

This paper considers the problem of finding a low rank matrix from observations of linear combinations of its elements. It is well known that if the problem fulfills a restricted isometry property (RIP), convex relaxations using the nuclear norm typically work well and come with theoretical performance guarantees. On the other hand these formulations suffer from a shrinking bias that can severely degrade the solution in the presence of noise. In this theoretical paper we study an alternative non-convex relaxation that in contrast to the nuclear norm does not penalize the leading singular values and thereby avoids this bias. We show that despite its non-convexity the proposed formulation will in many cases have a single stationary point if a RIP holds. Our numerical tests show that our approach typically converges to a better solution than nuclear norm based alternatives even in cases when the RIP does not hold.

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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
Proceedings - 2017 IEEE International Conference on Computer Vision Workshops, ICCVW 2017
volume
2018-January
pages
9 pages
publisher
Institute of Electrical and Electronics Engineers Inc.
conference name
16th IEEE International Conference on Computer Vision Workshops, ICCVW 2017
external identifiers
  • scopus:85046272238
ISBN
9781538610343
DOI
10.1109/ICCVW.2017.214
language
English
LU publication?
yes
id
4743450f-006b-4c43-86b0-f52e38ba18dc
date added to LUP
2018-05-17 15:05:42
date last changed
2018-05-29 11:18:04
@inproceedings{4743450f-006b-4c43-86b0-f52e38ba18dc,
  abstract     = {<p>This paper considers the problem of finding a low rank matrix from observations of linear combinations of its elements. It is well known that if the problem fulfills a restricted isometry property (RIP), convex relaxations using the nuclear norm typically work well and come with theoretical performance guarantees. On the other hand these formulations suffer from a shrinking bias that can severely degrade the solution in the presence of noise. In this theoretical paper we study an alternative non-convex relaxation that in contrast to the nuclear norm does not penalize the leading singular values and thereby avoids this bias. We show that despite its non-convexity the proposed formulation will in many cases have a single stationary point if a RIP holds. Our numerical tests show that our approach typically converges to a better solution than nuclear norm based alternatives even in cases when the RIP does not hold.</p>},
  author       = {Olsson, Carl and Carlsson, Marcus and Bylow, Erik},
  booktitle    = {Proceedings - 2017 IEEE International Conference on Computer Vision Workshops, ICCVW 2017},
  isbn         = {9781538610343},
  language     = {eng},
  month        = {01},
  pages        = {1809--1817},
  publisher    = {Institute of Electrical and Electronics Engineers Inc.},
  title        = {A Non-convex Relaxation for Fixed-Rank Approximation},
  url          = {http://dx.doi.org/10.1109/ICCVW.2017.214},
  volume       = {2018-January},
  year         = {2018},
}