A Combinatorial Approach to $L_1$-Matrix Factorization
(2015) In Journal of Mathematical Imaging and Vision 51(3). p.430-441- Abstract
- Recent work on low-rank matrix factorization has focused on the missing data problem and robustness to outliers and therefore the problem has often been studied under the $L_1$-norm. However, due to the non-convexity of the problem, most algorithms are sensitive to initialization and tend to get stuck in a local optimum.
In this paper, we present a new theoretical framework aimed at achieving optimal solutions to the factorization problem. We define a set of stationary points to the problem that will normally contain the optimal solution. It may be too time-consuming to check all these points, but we demonstrate on several practical applications that even by just computing a random subset of these stationary points, one... (More) - Recent work on low-rank matrix factorization has focused on the missing data problem and robustness to outliers and therefore the problem has often been studied under the $L_1$-norm. However, due to the non-convexity of the problem, most algorithms are sensitive to initialization and tend to get stuck in a local optimum.
In this paper, we present a new theoretical framework aimed at achieving optimal solutions to the factorization problem. We define a set of stationary points to the problem that will normally contain the optimal solution. It may be too time-consuming to check all these points, but we demonstrate on several practical applications that even by just computing a random subset of these stationary points, one can achieve significantly better results than current state of the art. In fact, in our experimental results we empirically observe that our competitors rarely find the optimal solution and that our approach is less sensitive to the existence of multiple local minima. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/4730646
- author
- Jiang, Fangyuan LU ; Enqvist, Olof and Kahl, Fredrik LU
- organization
- publishing date
- 2015
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- $L_1$-Matrix Factorization, Robust Estimation, Structure-from-Motion, Photometric Stereo
- in
- Journal of Mathematical Imaging and Vision
- volume
- 51
- issue
- 3
- pages
- 430 - 441
- publisher
- Springer
- external identifiers
-
- wos:000351540200007
- scopus:84925066132
- ISSN
- 0924-9907
- DOI
- 10.1007/s10851-014-0533-0
- language
- English
- LU publication?
- yes
- id
- 50803edf-5042-4ce1-b2ea-0cc3802a1391 (old id 4730646)
- date added to LUP
- 2016-04-01 10:14:03
- date last changed
- 2022-01-25 21:05:31
@article{50803edf-5042-4ce1-b2ea-0cc3802a1391,
abstract = {{Recent work on low-rank matrix factorization has focused on the missing data problem and robustness to outliers and therefore the problem has often been studied under the $L_1$-norm. However, due to the non-convexity of the problem, most algorithms are sensitive to initialization and tend to get stuck in a local optimum.<br/><br>
<br/><br>
In this paper, we present a new theoretical framework aimed at achieving optimal solutions to the factorization problem. We define a set of stationary points to the problem that will normally contain the optimal solution. It may be too time-consuming to check all these points, but we demonstrate on several practical applications that even by just computing a random subset of these stationary points, one can achieve significantly better results than current state of the art. In fact, in our experimental results we empirically observe that our competitors rarely find the optimal solution and that our approach is less sensitive to the existence of multiple local minima.}},
author = {{Jiang, Fangyuan and Enqvist, Olof and Kahl, Fredrik}},
issn = {{0924-9907}},
keywords = {{$L_1$-Matrix Factorization; Robust Estimation; Structure-from-Motion; Photometric Stereo}},
language = {{eng}},
number = {{3}},
pages = {{430--441}},
publisher = {{Springer}},
series = {{Journal of Mathematical Imaging and Vision}},
title = {{A Combinatorial Approach to $L_1$-Matrix Factorization}},
url = {{https://lup.lub.lu.se/search/files/1675450/4730650.pdf}},
doi = {{10.1007/s10851-014-0533-0}},
volume = {{51}},
year = {{2015}},
}