Accurate Optimization of Weighted Nuclear Norm for Non-Rigid Structure from Motion
(2020) 16th European Conference on Computer Vision, ECCV 2020 In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 12372 LNCS. p.21-37- Abstract
Fitting a matrix of a given rank to data in a least squares sense can be done very effectively using 2nd order methods such as Levenberg-Marquardt by explicitly optimizing over a bilinear parameterization of the matrix. In contrast, when applying more general singular value penalties, such as weighted nuclear norm priors, direct optimization over the elements of the matrix is typically used. Due to non-differentiability of the resulting objective function, first order sub-gradient or splitting methods are predominantly used. While these offer rapid iterations it is well known that they become inefficient near the minimum due to zig-zagging and in practice one is therefore often forced to settle for an approximate solution. In this paper... (More)
Fitting a matrix of a given rank to data in a least squares sense can be done very effectively using 2nd order methods such as Levenberg-Marquardt by explicitly optimizing over a bilinear parameterization of the matrix. In contrast, when applying more general singular value penalties, such as weighted nuclear norm priors, direct optimization over the elements of the matrix is typically used. Due to non-differentiability of the resulting objective function, first order sub-gradient or splitting methods are predominantly used. While these offer rapid iterations it is well known that they become inefficient near the minimum due to zig-zagging and in practice one is therefore often forced to settle for an approximate solution. In this paper we show that more accurate results can in many cases be achieved with 2nd order methods. Our main result shows how to construct bilinear formulations, for a general class of regularizers including weighted nuclear norm penalties, that are provably equivalent to the original problems. With these formulations the regularizing function becomes twice differentiable and 2nd order methods can be applied. We show experimentally, on a number of structure from motion problems, that our approach outperforms state-of-the-art methods.
(Less)
- author
- Iglesias, José Pedro ; Olsson, Carl LU and Valtonen Örnhag, Marcus LU
- organization
- publishing date
- 2020
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- host publication
- Computer Vision – ECCV 2020 - 16th European Conference, 2020, Proceedings
- series title
- Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
- editor
- Vedaldi, Andrea ; Bischof, Horst ; Brox, Thomas and Frahm, Jan-Michael
- volume
- 12372 LNCS
- pages
- 17 pages
- publisher
- Springer Science and Business Media B.V.
- conference name
- 16th European Conference on Computer Vision, ECCV 2020
- conference location
- Glasgow, United Kingdom
- conference dates
- 2020-08-23 - 2020-08-28
- external identifiers
-
- scopus:85097399749
- ISSN
- 0302-9743
- 1611-3349
- ISBN
- 9783030585822
- DOI
- 10.1007/978-3-030-58583-9_2
- language
- English
- LU publication?
- yes
- id
- bf2e3e7c-8c39-42db-b206-14105f3f038e
- date added to LUP
- 2020-12-22 12:22:34
- date last changed
- 2024-04-03 20:02:02
@inproceedings{bf2e3e7c-8c39-42db-b206-14105f3f038e,
abstract = {{<p>Fitting a matrix of a given rank to data in a least squares sense can be done very effectively using 2nd order methods such as Levenberg-Marquardt by explicitly optimizing over a bilinear parameterization of the matrix. In contrast, when applying more general singular value penalties, such as weighted nuclear norm priors, direct optimization over the elements of the matrix is typically used. Due to non-differentiability of the resulting objective function, first order sub-gradient or splitting methods are predominantly used. While these offer rapid iterations it is well known that they become inefficient near the minimum due to zig-zagging and in practice one is therefore often forced to settle for an approximate solution. In this paper we show that more accurate results can in many cases be achieved with 2nd order methods. Our main result shows how to construct bilinear formulations, for a general class of regularizers including weighted nuclear norm penalties, that are provably equivalent to the original problems. With these formulations the regularizing function becomes twice differentiable and 2nd order methods can be applied. We show experimentally, on a number of structure from motion problems, that our approach outperforms state-of-the-art methods.</p>}},
author = {{Iglesias, José Pedro and Olsson, Carl and Valtonen Örnhag, Marcus}},
booktitle = {{Computer Vision – ECCV 2020 - 16th European Conference, 2020, Proceedings}},
editor = {{Vedaldi, Andrea and Bischof, Horst and Brox, Thomas and Frahm, Jan-Michael}},
isbn = {{9783030585822}},
issn = {{0302-9743}},
language = {{eng}},
pages = {{21--37}},
publisher = {{Springer Science and Business Media B.V.}},
series = {{Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)}},
title = {{Accurate Optimization of Weighted Nuclear Norm for Non-Rigid Structure from Motion}},
url = {{http://dx.doi.org/10.1007/978-3-030-58583-9_2}},
doi = {{10.1007/978-3-030-58583-9_2}},
volume = {{12372 LNCS}},
year = {{2020}},
}