Abstract
The problem of missing data in multiway arrays (i.e., tensors) is common in many fields such as bibliographic data analysis, image processing, and computer vision. We consider the problems of approximating a tensor by another tensor with low multilinear rank in the presence of missing data and possibly reconstructing it (i.e., tensor completion). In this paper, we propose a weighted Tucker model which models only the known elements for capturing the latent structure of the data and reconstructing the missing elements. To treat the nonuniqueness of the proposed weighted Tucker model, a novel gradient descent algorithm based on a Grassmann manifold, which is termed Tucker weighted optimization (Tucker-Wopt), is proposed for guaranteeing the global convergence to a local minimum of the problem. Based on extensive experiments, Tucker-Wopt is shown to successfully reconstruct tensors with noise and up to 95% missing data. Furthermore, the experiments on traffic flow volume data demonstrate the usefulness of our algorithm on real-world application.
Highlights
Missing data can be recognized as the problem that occurs in the data collection process
We focus on low multilinear rank approximation of tensors with missing data based on Tucker decom
The differences between the two methods are that (1) we extend the matrix case to tensor case by laying out the theoretical foundations and build an efficient algorithm by using nonlinear optimization technique on Grassmann manifold and (2) that we focus on low multilinear rank approximation problem with missing data which is different and more complex than the matrix case
Summary
Missing data can be recognized as the problem that occurs in the data collection process. This problem is of considerable practical interest. It has been shown that data often have more than two modes of variation and are represented as multiway arrays (i.e., tensors). In internet network traffic flows analysis, the network traffic flows can be modeled as a fourth-order tensor with source IP, destination IP, port number, and time [1, 2]. Other examples include bibliographic data analysis [1, 2], image in-painting [3, 4], video in-painting [5], and data analysis [6]. It is necessary to develop a tool for reconstructing efficiently the tensors with missing data
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