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Minimizing the Maximal Rank

Bylow, Erik LU ; Olsson, Carl LU ; Kahl, Fredrik LU and Nilsson, Mikael LU (2016) 2016 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2016 In CVPR 2016, proceedings
Abstract
In computer vision, many problems can be formulated as finding a low rank approximation of a given matrix. Ideally, if all elements of the measurement matrix are available, this is easily solved in the L2-norm using factorization. However, in practice this is rarely the case. Lately, this problem has been addressed using different approaches, one is to replace the rank term by the convex nuclear norm, another is to derive the convex envelope of the rank term plus a data term. In the latter case, matrices are divided into sub-matrices and the envelope is computed for each sub-block individually. In this paper a new convex envelope is derived which takes all sub-matrices into account simultaneously. This leads to a simpler formulation, using... (More)
In computer vision, many problems can be formulated as finding a low rank approximation of a given matrix. Ideally, if all elements of the measurement matrix are available, this is easily solved in the L2-norm using factorization. However, in practice this is rarely the case. Lately, this problem has been addressed using different approaches, one is to replace the rank term by the convex nuclear norm, another is to derive the convex envelope of the rank term plus a data term. In the latter case, matrices are divided into sub-matrices and the envelope is computed for each sub-block individually. In this paper a new convex envelope is derived which takes all sub-matrices into account simultaneously. This leads to a simpler formulation, using only one parameter to control the trade-of between rank and data fit, for applications where one seeks low rank approximations of multiple matrices with the same rank. We show in this paper how our general framework can be used for manifold denoising of several images at once, as well as just denoising one image. Experimental comparisons show that our method achieves results similar to state-of-the-art approaches while being applicable for other problems such as linear shape model estimation.
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Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
CVPR 2016, proceedings
pages
9 pages
conference name
2016 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2016
external identifiers
  • scopus:84986260048
language
English
LU publication?
yes
id
a5fc8d48-430c-4110-aa31-4f0d7def5c15
date added to LUP
2016-09-08 11:50:15
date last changed
2017-01-01 08:33:35
@inproceedings{a5fc8d48-430c-4110-aa31-4f0d7def5c15,
  abstract     = {In computer vision, many problems can be formulated as finding a low rank approximation of a given matrix. Ideally, if all elements of the measurement matrix are available, this is easily solved in the L2-norm using factorization. However, in practice this is rarely the case. Lately, this problem has been addressed using different approaches, one is to replace the rank term by the convex nuclear norm, another is to derive the convex envelope of the rank term plus a data term. In the latter case, matrices are divided into sub-matrices and the envelope is computed for each sub-block individually. In this paper a new convex envelope is derived which takes all sub-matrices into account simultaneously. This leads to a simpler formulation, using only one parameter to control the trade-of between rank and data fit, for applications where one seeks low rank approximations of multiple matrices with the same rank. We show in this paper how our general framework can be used for manifold denoising of several images at once, as well as just denoising one image. Experimental comparisons show that our method achieves results similar to state-of-the-art approaches while being applicable for other problems such as linear shape model estimation.<br/>},
  author       = {Bylow, Erik and Olsson, Carl and Kahl, Fredrik and Nilsson, Mikael},
  booktitle    = {CVPR 2016, proceedings},
  language     = {eng},
  pages        = {9},
  title        = {Minimizing the Maximal Rank},
  year         = {2016},
}