Structured Low-Rank Matrix Factorization: Global Optimality, Algorithms, and Applications

Convex formulations of low-rank matrix factorization problems have received considerable attention in machine learning. However, such formulations often require solving for a matrix of the size of the data matrix, making it challenging to apply them to large scale datasets. Moreover, in many applica...

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Veröffentlicht in:IEEE transactions on pattern analysis and machine intelligence Jg. 42; H. 6; S. 1468 - 1482
Hauptverfasser: Haeffele, Benjamin D., Vidal, Rene
Format: Journal Article
Sprache:Englisch
Veröffentlicht: United States IEEE 01.06.2020
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Calcium
calcium imaging
Datasets
Factorization
hyperspectral compressed recovery
Image segmentation
Imaging
Low-rank matrix factorization
Machine learning
non-convex optimization
Optimization
Principal component analysis
Regularization
Video compression
Videos
ISSN:0162-8828, 1939-3539, 2160-9292, 1939-3539
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Zusammenfassung:Convex formulations of low-rank matrix factorization problems have received considerable attention in machine learning. However, such formulations often require solving for a matrix of the size of the data matrix, making it challenging to apply them to large scale datasets. Moreover, in many applications the data can display structures beyond simply being low-rank, e.g., images and videos present complex spatio-temporal structures that are largely ignored by standard low-rank methods. In this paper we study a matrix factorization technique that is suitable for large datasets and captures additional structure in the factors by using a particular form of regularization that includes well-known regularizers such as total variation and the nuclear norm as particular cases. Although the resulting optimization problem is non-convex, we show that if the size of the factors is large enough, under certain conditions, any local minimizer for the factors yields a global minimizer. A few practical algorithms are also provided to solve the matrix factorization problem, and bounds on the distance from a given approximate solution of the optimization problem to the global optimum are derived. Examples in neural calcium imaging video segmentation and hyperspectral compressed recovery show the advantages of our approach on high-dimensional datasets.
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ISSN:0162-8828
1939-3539
2160-9292
1939-3539
DOI:10.1109/TPAMI.2019.2900306