Sequential Unfolding SVD for Tensors With Applications in Array Signal Processing

Salmi, Jussi; Richter, Andreas; Koivunen, Visa

doi:10.1109/tsp.2009.2027740

Cited by 63 publications

(30 citation statements)

References 31 publications

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“…Sequential Unfolding SVD, PARATREE. The sequential unfolding SVD or PARATREE from [16] is defined quite similarly as the H-Tucker format. The first separation is via the SVD of the familiar form…”

Section: Comparison With Other Formatsmentioning

confidence: 99%

“…This new format allows the representation of order d tensors with (d − 1)k 3 + k d µ=1 n µ data, where k is the involved -implicitly defined -representation rank. A similar format has been presented by other groups: the tree Tucker and tensor train format [14,13] as well as the sequential unfolding SVD [16]. To our best knowledge the first successful approach to a hierarchical format has been developed by Beck & Jäckle & Worth & Meyer [1] and Wang & Thoss [18] (these references were kindly pointed out to us by Christian Lubich and Michael Griebel).…”

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confidence: 99%

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Hierarchical Singular Value Decomposition of Tensors

Grasedyck¹

2010

SIAM J. Matrix Anal. & Appl.

469

518

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Abstract. We define the hierarchical singular value decomposition (SVD) for tensors of order d ≥ 2. This hierarchical SVD has properties like the matrix SVD (and collapses to the SVD in d = 2), and we prove these. In particular, one can find low rank (almost) best approximations in a hierarchical format (H-Tucker) which requires only O((d − 1)k 3 + dnk) parameters, where d is the order of the tensor, n the size of the modes and k the (hierarchical) rank. The H-Tucker format is a specialization of the Tucker format and it contains as a special case all (canonical) rank k tensors. Based on this new concept of a hierarchical SVD we present algorithms for hierarchical tensor calculations allowing for a rigorous error analysis. The complexity of the truncation (finding lower rank approximations to hierarchical rank k tensors) is in O((d−1)k 4 +dnk 2 ) and the attainable accuracy is just 2-3 digits less than machine precision.

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Section: Comparison With Other Formatsmentioning

confidence: 99%

mentioning

confidence: 99%

Hierarchical Singular Value Decomposition of Tensors

Grasedyck¹

2010

SIAM J. Matrix Anal. & Appl.

469

518

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“…The block term decomposition (BTD) in rank-(1, L p , L p ) terms of a third-order tensor X ∈ C I×J×K , which is compared to a third-order PARATREE model in [78], can also be viewed as a particular CONFAC-(1,3) model. Indeed, such a decomposition can be written as [79] …”

Section: Paralind/confac-(n 1 N) Modelsmentioning

confidence: 99%

Overview of constrained PARAFAC models

Favier

Almeida

2014

EURASIP J. Adv. Signal Process.

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In this paper, we present an overview of constrained parallel factor (PARAFAC) models where the constraints model linear dependencies among columns of the factor matrices of the tensor decomposition or, alternatively, the pattern of interactions between different modes of the tensor which are captured by the equivalent core tensor. Some tensor prerequisites with a particular emphasis on mode combination using Kronecker products of canonical vectors that makes easier matricization operations, are first introduced. This Kronecker product-based approach is also formulated in terms of an index notation, which provides an original and concise formalism for both matricizing tensors and writing tensor models. Then, after a brief reminder of PARAFAC and Tucker models, two families of constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models, are described in a unified framework, for Nth-order tensors. New tensor models, called nested Tucker models and block PARALIND/CONFAC models, are also introduced. A link between PARATUCK models and constrained PARAFAC models is then established. Finally, new uniqueness properties of PARATUCK models are deduced from sufficient conditions for essential uniqueness of their associated constrained PARAFAC models.Keywords: Constrained PARAFAC; PARALIND/CONFAC; PARATUCK; Tensor models; Tucker models 1 Review IntroductionTensor calculus was introduced in differential geometry, at the end of the nineteenth century, and then tensor analysis was developed in the context of Einstein's theory of general relativity, with the introduction of index notation, the so-called Einstein summation convention, at the beginning of the twentieth century, which allows to simplify and shorten physics equations involving tensors. Index notation is also useful for simplifying multivariate statistical calculations, particularly those involving cumulant tensors [1]. Generally speaking, tensors are used in physics and differential geometry for characterizing the properties of a physical system, representing fundamental laws of physics, and defining geometrical objects whose components are functions. When these functions are defined over a continuum of points of a mathematical space, the tensor forms what is called a tensor field, a generalization of vector field used to solve problems involving curved surfaces or spaces, as it is the case of After the first tensor developments by mathematicians and physicists, the need of analyzing collections of data matrices that can be seen as three-way data arrays gave rise to three-way models for data analysis, with the pioneering works of Tucker in psychometrics [3], and Harshman in phonetics [4], who proposed what is now referred to as the Tucker and parallel factor (PARAFAC) decompositions, respectively. The PARAFAC decomposition was independently proposed by Carroll and Chang [5] under the name canonical decomposition (CANDE-COMP) and then called CANDECOMP/PARAFAC (CP) in [6]. For a history of the development of multi-way models in the context ...

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“…Due to orthogonality of the decomposition [16], the approximation error can be equivalently expressed in terms of the sum of the product of the singular values of the factors. These SVD SVD…”

Section: B Low Rank Susvd Approximationmentioning

confidence: 99%

Sequential unfolding SVD for low rank orthogonal tensor approximation

Salmi¹,

Richter²,

Koivunen³

2008

2008 42nd Asilomar Conference on Signals, Systems and Computers

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This paper contributes to the field of N-way (N ≥ 3) tensor decompositions, which are increasingly popular in various signal processing applications. A novel PARATREE decomposition structure is introduced, accompanied with Sequential Unfolding SVD (SUSVD) algorithm. SUSVD applies a matrix SVD sequentially on the unfolded tensor, which is reshaped from the right hand basis vectors of the SVD of the previous mode. The consequent PARATREE model is related to the well known family of PARAFAC tensor decompositions, describing a tensor as a sum of rank-1 tensors. PARATREE is an efficient model to be used for orthogonal lower rank approximations, offering significant computational savings in algorithm implementations due to a hierarchical tree structure. The performance of the proposed algorithm is illustrated through an application of measurement noise suppression in wideband MIMO measurements.

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Sequential Unfolding SVD for Tensors With Applications in Array Signal Processing

Cited by 63 publications

References 31 publications

Hierarchical Singular Value Decomposition of Tensors

Hierarchical Singular Value Decomposition of Tensors

Overview of constrained PARAFAC models

Sequential unfolding SVD for low rank orthogonal tensor approximation

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