Triple Decomposition and Tensor Recovery of Third Order Tensors

Qi, Liqun; Chen, Yannan; Bakshi, Mayank; Zhang, Xinzhen

doi:10.1137/20m1323266

Cited by 20 publications

(16 citation statements)

References 15 publications

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“…In this section, we briefly overview some basic definitions, including NTD [24,26,27], TriD [10,18], hypergraph learning [28][29][30]. The notations used in this paper are listed in Table 1.…”

Section: Related Workmentioning

confidence: 99%

“…In the TD and NTD methods, the size of the core tensor grows rapidly as the order of data increases, which may result in a high cost of calculation. To overcome this shortcoming, Qi et al [10] recently proposed a new form of triple decomposition for third-order tensors, which reduces a third-order tensor to the product of three thirdorder factor tensors. Definition 1.…”

Section: Bilevel Form Of Triple Decomposition (Trid)mentioning

confidence: 99%

“…Definition 2. [10] Based on the definition of NTD shown in (1), if the core tensor X has a triple decomposition X = 〚ABC〛, where TriRank( X )=r, A ∈ R r1×r×r + , B ∈ R r×r2×r + , and C ∈ R r×r×r3 + . Then X can be represented as…”

Section: Bilevel Form Of Triple Decomposition (Trid)mentioning

confidence: 99%

“…This variability is widespread in some real data, such as traffic and internet data, where the three modes of the third-order tensor have strong temporal, spatial, and periodic significance [9]. To remedy these shortcomings, Qi et al [10] proposed a bilevel form of triple decomposition (TriD). The triple decomposition for third-order tensors transforms a third-order tensor into a product of three third-order factor tensors.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Hypergraph regularized nonnegative triple decomposition for multiway data analysis

Liao,

Liu,

Razak

2024

Preprint

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Tucker decomposition is widely used for image representation, data reconstruction , and machine learning tasks, but the calculation cost for updating the Tucker core is high. Bilevel form of triple decomposition (TriD) overcomes this issue by decomposing the Tucker core into three low-dimensional third-order factor tensors and plays an important role in the dimension reduction of data representation. TriD, on the other hand, is incapable of precisely encoding similarity relationships for tensor data with a complex manifold structure. To address this shortcoming, we take advantage of hypergraph learning and propose a novel hypergraph regu-larized nonnegative triple decomposition for multiway data analysis that employs the hypergraph to model the complex relationships among the raw data. Furthermore , we develop a multiplicative update algorithm to solve our optimization problem and theoretically prove its convergence. Finally, we perform extensive numerical tests on six real-world datasets, and the results show that our proposed algorithm outperforms some state-of-the-art methods.

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Section: Related Workmentioning

confidence: 99%

Section: Bilevel Form Of Triple Decomposition (Trid)mentioning

confidence: 99%

Section: Bilevel Form Of Triple Decomposition (Trid)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hypergraph regularized nonnegative triple decomposition for multiway data analysis

Liao,

Liu,

Razak

2024

Preprint

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“…As a higher order generalization of matrix completion, tensor completion has attracted much more attention recently [25], [26], [27], [28], [29], [30]. Compared to matrix rank, there are various definitions for tensor rank, including CAN-DECOMP/PARAFAC (CP) rank [31], Tucker rank [32], TT rank [33], triple rank [34] and tubal rank [35]. Since it is generally NP-hard to compute the CP rank [36], it is hard to apply CP rank to the tensor completion problem.…”

Section: Introductionmentioning

confidence: 99%

Tensor factorization based method for low rank matrix completion and its application on tensor completion

Yu¹,

Zhang²

2022

Preprint

Self Cite

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Low rank matrix and tensor completion problems are to recover the incomplete two and higher order data by using their low rank structures. The essential problem in the matrix and tensor completion problems is how to improve the efficiency. To this end, we first establish the relationship between matrix rank and tensor tubal rank, and then reformulate matrix completion problem as a tensor completion problem. For the reformulated tensor completion problem, we adopt a two-stage strategy based on tensor factorization algorithm. In this way, a matrix completion problem of big size can be solved via some matrix computations of smaller sizes. For a third order tensor completion problem, to fully exploit the low rank structures, we introduce the double tubal rank which combines the tubal rank and the rank of the mode-3 unfolding matrix. For the mode-3 unfolding matrix rank, we follow the idea of matrix completion. Based on this, we establish a novel model and modify the tensor factorization based algorithm for third order tensor completion. Extensive numerical experiments demonstrate that the proposed methods outperform state-of-the-art methods in terms of both accuracy and running time.

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