The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP-ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP-ALS subproblems efficiently, have the same complexity as the standard CP-ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill-conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.
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Accurate Tensor Decomposition with Simultaneous Rank Approximation for Surveillance Videos
Canonical polyadic (CP) decomposition of a tensor is a basic operation in a lot of applications such as data mining and video foreground/background separation. However, existing algorithms for CP decomposition require users to provide a rank of the target tensor data as part of the input and finding the rank of a tensor is an NP-hard problem. Currently, to perform CP decomposition, users are required to make an informed guess of a proper tensor rank based on the data at hand, and the result may still be suboptimal. In this paper, we propose to conduct CP decomposition and tensor rank approximation together, so that users do not have to provide the proper rank beforehand, and the decomposition algorithm will find the proper rank and return a high-quality result. We formulate an optimization problem with an objective function consisting of a least-squares Tikhonov regularization and a sparse L1-regularization term. We also test its effectiveness over real applications with moving object videos.
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- Award ID(s):
- 1755464
- NSF-PAR ID:
- 10331956
- Date Published:
- Journal Name:
- 54th Asilomar Conference on Signals, Systems, and Computers
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/IEEECONF51394.2020.9443285
