Multi-Way Compressed Sensing for Sparse Low-Rank Tensors

Sidiropoulos, Nicholas D.; Kyrillidis, Anastasios

doi:10.1109/lsp.2012.2210872

Cited by 124 publications

(81 citation statements)

References 29 publications

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“…Many extensions of these methods to tensors have been developed [10], [22] and new methods tailored to tensors have emerged, e.g. [23], [24]. In this tutorial we focus on a class of CS methods where decompositions of very large tensors are computed using only a small number of known elements.…”

Section: Computing Decompositions Of Large Incomplete Tensorsmentioning

confidence: 99%

Breaking the Curse of Dimensionality Using Decompositions of Incomplete Tensors: Tensor-based scientific computing in big data analysis

Vervliet

Debals

Sorber

et al. 2014

IEEE Signal Process. Mag.

148

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Higher-order tensors and their decompositions are abundantly present in domains such as signal processing (e.g. higher-order statistics [1], sensor array processing [2]), scientific computing (e.g. discretized multivariate functions [3]- [6]) and quantum information theory (e.g. representation of quantum many-body states [7]). In many applications the, possibly huge, tensors can be approximated well by compact multilinear models or decompositions. Tensor decompositions are more versatile tools than the linear models resulting from traditional matrix approaches.Compared to matrices, tensors have at least one extra dimension. The number of elements in a tensor increases exponentially with the number of dimensions, and so do the computational and memory requirements. The exponential dependency (and the problems that are caused by it) is called the curse of dimensionality. The curse limits the order of the tensors that can be handled.Even for modest order, tensor problems are often large-scale. Large tensors can be handled, and the curse can be alleviated or even removed, by using a decomposition that represents the tensor, instead of the tensor itself. However, most decomposition algorithms require full tensors, which renders these algorithms infeasible for large datasets. If a tensor can be represented by a decomposition, this hypothesized structure can be exploited by using compressed sensing type methods working on incomplete tensors, i.e. tensors with only a few known elements.In domains such as scientific computing and quantum information theory, tensor decompositions such as the Tucker decomposition and tensor trains have successfully been applied to represent large tensors. In the latter case, the tensor can contain more elements than the number of atoms in the universe [8]

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Section: Computing Decompositions Of Large Incomplete Tensorsmentioning

confidence: 99%

Breaking the Curse of Dimensionality Using Decompositions of Incomplete Tensors: Tensor-based scientific computing in big data analysis

Vervliet

Debals

Sorber

et al. 2014

IEEE Signal Process. Mag.

148

View full text Add to dashboard Cite

show abstract

“…As massive amounts of data often lead to limitations and challenges in analysis, sparsity is often imposed on the latent factors to improve the analysis and inference learning [13]. Mathematically, the nonnegative sparse tensor decomposition is formulated as follows [12],…”

Section: Sparsity Analysis Of Latent Factors In Matrix/tensor Decompomentioning

confidence: 99%

Parameter selection for nonnegativel1matrix/tensor sparse decomposition

Wang

Liu

Caccetta

et al. 2015

Operations Research Letters

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“…In 2011, Duarte and Eldar introduced the Kronecker structure into dictionary constructing, proposing the Kronecker-CS framework [26]. In 2012, Sidiropoulos and Kyrillidis investigated CS theorem with regard to low-rank tensor signal processing, proposing a two-step sparse reconstruction algorithm [27]. In 2013, Caiafa and Cichocki proposed two types of matching pursuit algorithms based on tensor decomposition [28].…”

Section: Introductionmentioning

confidence: 99%

Multi-linear sparse reconstruction for SAR imaging based on higher-order SVD

Gao

Gui

Cong

et al. 2017

EURASIP J. Adv. Signal Process.

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This paper focuses on the spotlight synthetic aperture radar (SAR) imaging for point scattering targets based on tensor modeling. In a real-world scenario, scatterers usually distribute in the block sparse pattern. Such a distribution feature has been scarcely utilized by the previous studies of SAR imaging. Our work takes advantage of this structure property of the target scene, constructing a multi-linear sparse reconstruction algorithm for SAR imaging. The multi-linear block sparsity is introduced into higher-order singular value decomposition (SVD) with a dictionary constructing procedure by this research. The simulation experiments for ideal point targets show the robustness of the proposed algorithm to the noise and sidelobe disturbance which always influence the imaging quality of the conventional methods. The computational resources requirement is further investigated in this paper. As a consequence of the algorithm complexity analysis, the present method possesses the superiority on resource consumption compared with the classic matching pursuit method. The imaging implementations for practical measured data also demonstrate the effectiveness of the algorithm developed in this paper.

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Multi-Way Compressed Sensing for Sparse Low-Rank Tensors

Cited by 124 publications

References 29 publications

Breaking the Curse of Dimensionality Using Decompositions of Incomplete Tensors: Tensor-based scientific computing in big data analysis

Breaking the Curse of Dimensionality Using Decompositions of Incomplete Tensors: Tensor-based scientific computing in big data analysis

Parameter selection for nonnegativel1matrix/tensor sparse decomposition

Multi-linear sparse reconstruction for SAR imaging based on higher-order SVD

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