A Parallelizable Method for Missing Internet Traffic Tensor Data

Chen, Ling; Yu, Gaohang; Qi, Liqun; Xu, Yanwei

doi:10.48550/arxiv.2005.09838

Cited by 4 publications

(4 citation statements)

References 53 publications

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“…T-product and T-SVD Kilmer and her collaborators proposed T-product and T-SVD factorization of third order tensors [6,7,8,14,21,22,23]. Works on applications of T-product and T-SVD factorization include [3,9,11,15,18,19,20,24]. It is shown that they are very useful in applications.…”

Section: Related Workmentioning

confidence: 99%

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T-Singular Values and T-Sketching for Third Order Tensors

Qi,

2021

Preprint

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Based upon the T-SVD (tensor SVD) of third order tensors, introduced by Kilmer and her collaborators, we define T-singular values of third order tensors. T-singular values of third order tensors are nonnegative scalars. The number of nonzero T-singular values is the tensor tubal rank of the tensor. We then use T-singular values to define the tail energy of a third order tensor, and apply it to the error estimation of a tensor sketching algorithm for low rank tensor approximation. Numerical experiments on real world data show that our algorithm is efficient.

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Section: Related Workmentioning

confidence: 99%

“…The T-product operation, T-SVD decomposition and tensor tubal ranks were introduced by Kilmer and her collaborators in [6,7,8,23]. It is now widely used in engineering [3,9,14,15,18,19,20,21,22,24].…”

Section: Tensor-tensor Product Operationsmentioning

confidence: 99%

T-Singular Values and T-Sketching for Third Order Tensors

Qi,

2021

Preprint

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“…The T-product operation, T-SVD factorization and tensor tubal ranks were introduced by Kilmer and her collaborators in [3,4,5,18]. They are now widely used in engineering [1,7,11,12,13,14,15,16,17,19,20]. In particular, Kilmer and Martin [4] proposed T-SVD factorization.…”

Section: Introductionmentioning

confidence: 99%

An Orthogonal Equivalence Theorem for Third Order Tensors

Chen

Liu

et al. 2021

Preprint

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In 2011, Kilmer and Martin proposed tensor singular value decomposition (T-SVD) for third order tensors. Since then, T-SVD has applications in low rank tensor approximation, tensor recovery, multi-view clustering, multi-view feature extraction, tensor sketching, etc. By going through the Discrete Fourier Transform (DFT), matrix SVD and inverse DFT, a third order tensor is mapped to an f-diagonal third order tensor. We call this a Kilmer-Martin mapping. We show that the Kilmer-Martin mapping of a third order tensor is invariant if that third order tensor is taking T-product with some orthogonal tensors. We define singular values and T-rank of that third order tensor based upon its Kilmer-Martin mapping. Thus, tensor tubal rank, T-rank, singular values and T-singular values of a third order tensor are invariant when it is taking T-product with some orthogonal tensors. Some properties of singular values, T-rank and best T-rank one approximation are discussed.

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“…The matrix SVD factorization (1.1) was extended to third order tensors as T-SVD factorization by Kilmer and Martin [4,5]. The T-SVD factorization has been found wide applications in engineering and tensor computation [1,3,6,7,8,10,11,13,14,15,16,17,18,19,21,20,22,23]. In T-SVD factorization, an m × n × p third order real tensor A is decomposed to the tensor product of an m × m × p orthogonal tensor U, an m × n × p real f-diagonal tensor S, and an n × n × p orthogonal tensor V:…”

Section: Introductionmentioning

confidence: 99%

ST-SVD Factorization and s-Diagonal Tensors

Chen¹,

Liu²,

Chen³

et al. 2021

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A third order real tensor is mapped to a special f-diagonal tensor by going through Discrete Fourier Transform (DFT), standard matrix SVD and inverse DFT. We call such an f-diagonal tensor an s-diagonal tensor. An f-diagonal tensor is an s-diagonal tensor if and only if it is mapped to itself in the above process. The third order tensor space is partitioned to orthogonal equivalence classes. Each orthogonal equivalence class has a unique s-diagonal tensor. Two s-diagonal tensors are equal if they are orthogonally equivalent. Third order tensors in an orthogonal equivalence class have the same tensor tubal rank and T-singular values. Four meaningful necessary conditions for s-diagonal tensors are presented. Then we present a set of sufficient and necessary conditions for sdiagonal tensors. Such conditions involve a special complex number. In the cases that the dimension of the third mode is 2, 3 and 4, we present direct sufficient and necessary conditions which do not involve such a complex number.

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A Parallelizable Method for Missing Internet Traffic Tensor Data

Cited by 4 publications

References 53 publications

T-Singular Values and T-Sketching for Third Order Tensors

T-Singular Values and T-Sketching for Third Order Tensors

An Orthogonal Equivalence Theorem for Third Order Tensors

ST-SVD Factorization and s-Diagonal Tensors

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