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Total variation based tensor decomposition for multi‐dimensional data with time dimension
Author(s) -
Chen Chuan,
Li Xutao,
Ng Michael K.,
Yuan Xiaoming
Publication year - 2015
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1993
Subject(s) - piecewise , pixel , total variation denoising , mathematics , dimension (graph theory) , set (abstract data type) , algorithm , regularization (linguistics) , convergence (economics) , tensor (intrinsic definition) , variation (astronomy) , computer science , mathematical optimization , image (mathematics) , artificial intelligence , mathematical analysis , combinatorics , physics , astrophysics , economics , programming language , economic growth , pure mathematics
Summary In this paper, we study tensors with time dimension which arises in many scientific and engineering applications such as time series gene expression analysis and video analysis. In these applications, we are interested in determining a set of components interacting closely and consistently together over a period of time. The main aim of this paper is to develop a numerical method to compute such constrained CANDECOMP/PARAFAC (CP) decompositions. We make use of the total variation regularization to constrain the time dimension factor in the decomposition in order to obtain a piecewise constant function with respect to time points. The components of the other dimensions corresponding to these time points are closely related. For example, in time series gene expression analysis, a set of genes may regulate a biological process together during a specific time period; in video analysis, a set of image pixels may refer to a foreground object in the video frames. We employ ADMM to solve the resulting optimization problem, and study its convergence. Numerical examples on synthetic and real data sets are used to demonstrate that the proposed total variation based CP decomposition model can provide more accurate and interesting results. Copyright © 2015 John Wiley & Sons, Ltd.