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J. Liu,  P. Musialski,  P. Wonka,  J. Ye (2009)

Tensor Completion for Estimating Missing Values in Visual Data

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Proceedings of International Conference on Computer Vision (ICCV 2009)

Abstract

In this paper we propose an algorithm to estimate missing values in tensors of visual data. The values can be missing due to problems in the acquisition process, or because the user manually identi{fi}ed unwanted outliers. Our algorithm works even with a small amount of samples and it can propagate structure to {fi}ll larger missing regions. Our methodology is built on recent studies about matrix completion using the matrix trace norm. The contribution of our paper is to extend the matrix case to the tensor case by laying out the theoretical foundations and then by building a working algorithm. First, we propose a de{fi}nition for the tensor trace norm, that extends the established de{fi}nition of the matrix trace norm. Second, similar to matrix completion, the tensor completion is formulated as a convex optimization problem. Unfortunately, the straightforward problem extension is signi{fi}cantly harder to solve than the matrix case because of the dependency among multiple constraints. To tackle this problem, we employ a relaxation technique to separate the dependant relationships and use the block coordinate descent (BCD) method to achieve a globally optimal solution. Our experiments show potential applications of our algorithm and the quantitative evaluation indicates that our method is more accurate and robust than heuristic approaches.

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