Structured Low-Rank Matrix Factorization: Optimality, Algorithm, and Applications to Image Processing

Benjamin Haeffele, Eric Young, Rene Vidal
Proceedings of the 31st International Conference on Machine Learning, PMLR 32(2):2007-2015, 2014.

Abstract

Recently, convex solutions to low-rank matrix factorization problems have received increasing attention in machine learning. However, in many applications the data can display other structures beyond simply being low-rank. For example, images and videos present complex spatio-temporal structures, which are largely ignored by current low-rank methods. In this paper we explore a matrix factorization technique suitable for large datasets that captures additional structure in the factors by using a projective tensor norm, which includes classical image regularizers such as total variation and the nuclear norm as particular cases. Although the resulting optimization problem is not convex, we show that under certain conditions on the factors, any local minimizer for the factors yields a global minimizer for their product. Examples in biomedical video segmentation and hyperspectral compressed recovery show the advantages of our approach on high-dimensional datasets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v32-haeffele14, title = {Structured Low-Rank Matrix Factorization: Optimality, Algorithm, and Applications to Image Processing}, author = {Haeffele, Benjamin and Young, Eric and Vidal, Rene}, booktitle = {Proceedings of the 31st International Conference on Machine Learning}, pages = {2007--2015}, year = {2014}, editor = {Xing, Eric P. and Jebara, Tony}, volume = {32}, number = {2}, series = {Proceedings of Machine Learning Research}, address = {Bejing, China}, month = {22--24 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v32/haeffele14.pdf}, url = {https://proceedings.mlr.press/v32/haeffele14.html}, abstract = {Recently, convex solutions to low-rank matrix factorization problems have received increasing attention in machine learning. However, in many applications the data can display other structures beyond simply being low-rank. For example, images and videos present complex spatio-temporal structures, which are largely ignored by current low-rank methods. In this paper we explore a matrix factorization technique suitable for large datasets that captures additional structure in the factors by using a projective tensor norm, which includes classical image regularizers such as total variation and the nuclear norm as particular cases. Although the resulting optimization problem is not convex, we show that under certain conditions on the factors, any local minimizer for the factors yields a global minimizer for their product. Examples in biomedical video segmentation and hyperspectral compressed recovery show the advantages of our approach on high-dimensional datasets.} }
Endnote
%0 Conference Paper %T Structured Low-Rank Matrix Factorization: Optimality, Algorithm, and Applications to Image Processing %A Benjamin Haeffele %A Eric Young %A Rene Vidal %B Proceedings of the 31st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2014 %E Eric P. Xing %E Tony Jebara %F pmlr-v32-haeffele14 %I PMLR %P 2007--2015 %U https://proceedings.mlr.press/v32/haeffele14.html %V 32 %N 2 %X Recently, convex solutions to low-rank matrix factorization problems have received increasing attention in machine learning. However, in many applications the data can display other structures beyond simply being low-rank. For example, images and videos present complex spatio-temporal structures, which are largely ignored by current low-rank methods. In this paper we explore a matrix factorization technique suitable for large datasets that captures additional structure in the factors by using a projective tensor norm, which includes classical image regularizers such as total variation and the nuclear norm as particular cases. Although the resulting optimization problem is not convex, we show that under certain conditions on the factors, any local minimizer for the factors yields a global minimizer for their product. Examples in biomedical video segmentation and hyperspectral compressed recovery show the advantages of our approach on high-dimensional datasets.
RIS
TY - CPAPER TI - Structured Low-Rank Matrix Factorization: Optimality, Algorithm, and Applications to Image Processing AU - Benjamin Haeffele AU - Eric Young AU - Rene Vidal BT - Proceedings of the 31st International Conference on Machine Learning DA - 2014/06/18 ED - Eric P. Xing ED - Tony Jebara ID - pmlr-v32-haeffele14 PB - PMLR DP - Proceedings of Machine Learning Research VL - 32 IS - 2 SP - 2007 EP - 2015 L1 - http://proceedings.mlr.press/v32/haeffele14.pdf UR - https://proceedings.mlr.press/v32/haeffele14.html AB - Recently, convex solutions to low-rank matrix factorization problems have received increasing attention in machine learning. However, in many applications the data can display other structures beyond simply being low-rank. For example, images and videos present complex spatio-temporal structures, which are largely ignored by current low-rank methods. In this paper we explore a matrix factorization technique suitable for large datasets that captures additional structure in the factors by using a projective tensor norm, which includes classical image regularizers such as total variation and the nuclear norm as particular cases. Although the resulting optimization problem is not convex, we show that under certain conditions on the factors, any local minimizer for the factors yields a global minimizer for their product. Examples in biomedical video segmentation and hyperspectral compressed recovery show the advantages of our approach on high-dimensional datasets. ER -
APA
Haeffele, B., Young, E. & Vidal, R.. (2014). Structured Low-Rank Matrix Factorization: Optimality, Algorithm, and Applications to Image Processing. Proceedings of the 31st International Conference on Machine Learning, in Proceedings of Machine Learning Research 32(2):2007-2015 Available from https://proceedings.mlr.press/v32/haeffele14.html.

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