Smooth PARAFAC Decomposition for Tensor Completion

Yokota, Tatsuya; Zhao, Qibin; Cichocki, Andrzej

doi:10.1109/tsp.2016.2586759

Cited by 243 publications

(167 citation statements)

References 68 publications

(120 reference statements)

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“…We compared the performance of the proposed method with those of state-of-the-art tensor completion algorithms: HaLRTC (nuclear-norm regularization) [16], TV regularization [32], nuclear-norm and TV regularization (LR&TV) [28], STDC (constrained Tucker decomposition) [4], and SPCQV (constrained PARAFAC tensor decomposition) [30]. We prepared six missing images for this experiment.…”

Section: Comparison Using Color Imagesmentioning

confidence: 99%

See 1 more Smart Citation

Missing Slice Recovery for Tensors Using a Low-Rank Model in Embedded Space

Yokota

Erem

Güler

et al. 2018

2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition

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Let us consider a case where all of the elements in some continuous slices are missing in tensor data. In this case, the nuclear-norm and total variation regularization methods usually fail to recover the missing elements. The key problem is capturing some delay/shift-invariant structure. In this study, we consider a low-rank model in an embedded space of a tensor. For this purpose, we extend a delay embedding for a time series to a "multi-way delay-embedding transform" for a tensor, which takes a given incomplete tensor as the input and outputs a higher-order incomplete Hankel tensor. The higher-order tensor is then recovered by Tucker-based low-rank tensor factorization. Finally, an estimated tensor can be obtained by using the inverse multiway delay embedding transform of the recovered higherorder tensor. Our experiments showed that the proposed method successfully recovered missing slices for some color images and functional magnetic resonance images.

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Section: Comparison Using Color Imagesmentioning

confidence: 99%

“…Matrix/tensor completion is a technique for recovering the missing elements in incomplete data and it has become a very important method in recent years [2,3,1,9,16,21,4,31,30]. In general, completion is an ill-posed problem without any assumptions.…”

Section: Introductionmentioning

confidence: 99%

Missing Slice Recovery for Tensors Using a Low-Rank Model in Embedded Space

Yokota

Erem

Güler

et al. 2018

2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition

Self Cite

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“…The problem of estimating missing entries in higher dimensional arrays has been widely studied in recent years and there is a long list of available algorithms to choose from [22,20,27,28,25,29,24,30,31] [32,33,34,35,36]. Tensor completion has been recently proposed for handling missing measurements before classification, for example, in human activity recognition [37,38].…”

Section: Tensor Completion Algorithmsmentioning

confidence: 99%

Brain-Computer Interface with Corrupted EEG Data: a Tensor Completion Approach

et al. 2018

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Background / Introduction: One of the current issues in Brain-Computer Interface (BCI) is how to deal with noisy Electroencephalography (EEG) measurements organized as multidimensional datasets (tensors). On the other hand, recently, significant advances have been made in multidimensional signal completion algorithms that exploit tensor decomposition models to capture the intricate relationship among entries in a multidimensional signal. We propose to use tensor completion applied to EEG data for improving the classification performance in a motor imagery BCI system with corrupted measurements. Noisy measurements (electrode misconnections, subject movements, etc.) are considered as unknowns (missing samples) that are inferred from a tensor decomposition model (tensor completion). Methods: We evaluate the performance of four recently proposed tensor completion algorithms, CP-WOPT (Acar et al, 2011), 3DPB-TC (Caiafa et al, 2013), BCPF (Zhao et al, 2015) and HaLRT (Liu et al. ( 2013), plus a simple interpolation strategy, first with random missing entries and then with missing samples constrained to have a specific structure (random missing channels), which is a more realistic assumption in BCI applications. Results:We measured the ability of these algorithms to reconstruct the tensor from observed data.Then, we tested the classification accuracy of imagined movement in a BCI experiment with missing samples. We show that for random missing entries, all tensor completion algorithms can recover missing samples increasing the classification performance compared to a simple interpolation approach. For the random missing channels case, we show that tensor completion algorithms help to reconstruct missing channels, significantly improving the accuracy in the classification of motor imagery (MI), however, not at the same level as clean data. Summarizing, compared to the interpolation case, all tensor completion algorithms succeed to increase the classification performance by 7 -9% (LDA -SVD) for random missing entries and 15 -8% (LDA -SVD) for random missing channels.Conclusions: Tensor completion algorithms are useful in real BCI applications. The proposed strategy could allow using motor imagery BCI systems even when EEG data is highly affected by missing channels and/or samples, avoiding the need of new acquisitions in the calibration stage.

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“…Tensor completion has been widely studied in the literature. Various approaches like decomposition-based methods [29][30][31][32][33][34] are available in the literature. Various other approaches can be found in other works.…”

Section: Introductionmentioning

confidence: 99%

Quantized CP approximation and sparse tensor interpolation of function‐generated data

Khoromskij

Naraparaju

Schneider

2019

Numerical Linear Algebra App

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Summary In this article, we consider the iterative schemes to compute the canonical polyadic (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research on the quantics‐tensor train (QTT) method (“O(d log N)‐quantics approximation of N‐d tensors in high‐dimensional numerical modeling” in Constructive Approximation, 2011) developed for the tensor train (TT) approximation of the quantized images of function related data. In the QTT approach, the target vector of length 2L is reshaped to a Lth‐order tensor with two entries in each mode (quantized representation) and then approximated by the QTT tensor including 2r2L parameters, where r is the maximal TT rank. In what follows, we consider the alternating least squares (ALS) iterative scheme to compute the rank‐r CP approximation of the quantized vectors, which requires only 2rL≪2L parameters for storage. In the earlier papers (“Tensors‐structured numerical methods in scientific computing: survey on recent advances” in Chemom Intell Lab Syst, 2012), such a representation was called QCan format, whereas in this paper, we abbreviate it as the QCP (quantized canonical polyadic) representation. We test the ALS algorithm to calculate the QCP approximation on various functions, and in all cases, we observed the exponential error decay in the QCP rank. The main idea for recovering a discretized function in the rank‐r QCP format using the reduced number of the functional samples, calculated only at O(2rL) grid points, is presented. The special version of the ALS scheme for solving the arising minimization problem is described. This approach can be viewed as the sparse QCP‐interpolation method that allows to recover all 2rL representation parameters of the rank‐r QCP tensor. Numerical examples show the efficiency of the QCP‐ALS‐type iteration and indicate the exponential convergence rate in r.

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Smooth PARAFAC Decomposition for Tensor Completion

Cited by 243 publications

References 68 publications

Missing Slice Recovery for Tensors Using a Low-Rank Model in Embedded Space

Missing Slice Recovery for Tensors Using a Low-Rank Model in Embedded Space

Brain-Computer Interface with Corrupted EEG Data: a Tensor Completion Approach

Quantized CP approximation and sparse tensor interpolation of function‐generated data

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