Hierarchical Singular Value Decomposition of Tensors

Grasedyck, Lars

doi:10.1137/090764189

Cited by 469 publications

(531 citation statements)

References 13 publications

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“…Then, the domain of an ndimensional problem with, for example, n being some power of two, can be split into the tensor product of two domains of dimension n/2 which can be recursively further split until a terminal situation (a onedimensional domain or a truly higher-dimensional but nontensor-product domain) is reached. Related representation methods have recently been considered in Hackbusch & Kühn (2009) or Oseledets & Tyrtyshnikov (2009), Grasedyck (2010), Bebendorf (2011), Hackbusch (2012.…”

Section: Introductionmentioning

confidence: 99%

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Griebel

Harbrecht

2013

IMA Journal of Numerical Analysis

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We compare the cost complexities of two approximation schemes for functions f ∈ H p (Ω 1 × Ω 2 ) which live on the product domain Ω 1 × Ω 2 of sufficiently smooth domains Ω 1 ⊂ R n 1 and Ω 2 ⊂ R n 2 , namely the singular value/Karhunen-Lòeve decomposition and the sparse grid representation. Here, we assume that suitable finite element methods with associated fixed order r of accuracy are given on the domains Ω 1 and Ω 2 . Then, the sparse grid approximation essentially needs only O(ε −q ), with q = max{n 1 , n 2 }/r, unknowns to reach a prescribed accuracy ε, provided that the smoothness of f satisfies p r((n 1 + n 2 )/max{n 1 , n 2 }), which is an almost optimal rate. The singular value decomposition produces this rate only if f is analytical, since otherwise the decay of the singular values is not fast enough. If p < r((n 1 + n 2 )/max{n 1 , n 2 }), then the sparse grid approach gives essentially the rate O(ε −q ) with q = (n 1 + n 2 )/p, while, for the singular value decomposition, we can only prove the rate O(ε −q ) with q = (2 min{r, p} min{n 1 , n 2 } + 2p max{n 1 , n 2 })/(2p − min{n 1 , n 2 }) min{r, p}. We derive the resulting complexities, compare the two approaches and present numerical results which demonstrate that these rates are also achieved in numerical practice.

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Section: Introductionmentioning

confidence: 99%

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Griebel

Harbrecht

2013

IMA Journal of Numerical Analysis

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“…Technically, this is achieved by employing the hierarchical singular value decomposition 53 to represent the PES in the hierarchical tensor format 27 (Section IV B). This procedure not only yields a near-optimal fit, its accuracy can also easily be estimated on the fly.…”

Section: Discussionmentioning

confidence: 99%

“…The resulting method has been described previously in the mathematical literature by Grasedyck as hierarchical SVD with leaves-to-root truncation (Algorithm 2 in Ref. 53). A similar technique has been used in Ref.…”

Section: B the Multi-layer Generalization Of Potfitmentioning

confidence: 99%

Multi-layer Potfit: An accurate potential representation for efficient high-dimensional quantum dynamics

Otto

2014

The Journal of Chemical Physics

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The multi-layer multi-configuration time-dependent Hartree method (ML-MCTDH) is a highly efficient scheme for studying the dynamics of high-dimensional quantum systems. Its use is greatly facilitated if the Hamiltonian of the system possesses a particular structure through which the multi-dimensional matrix elements can be computed efficiently. In the field of quantum molecular dynamics, the effective interaction between the atoms is often described by potential energy surfaces (PES), and it is necessary to fit such PES into the desired structure. For high-dimensional systems, the current approaches for this fitting process either lead to fits that are too large to be practical, or their accuracy is difficult to predict and control.This article introduces multi-layer Potfit (MLPF), a novel fitting scheme that results in a PES representation in the hierarchical tensor (HT) format. The scheme is based on the hierarchical singular value decomposition, which can yield a near-optimal fit and give strict bounds for the obtained accuracy. Here, a recursive scheme for using the HT-format PES within ML-MCTDH is derived, and theoretical estimates as well as a computational example show that the use of MLPF can reduce the numerical effort for ML-MCTDH by orders of magnitude, compared to the traditionally used Potfit representation of the PES. Moreover, it is shown that MLPF is especially beneficial for high-accuracy PES representations, and it turns out that MLPF leads to computational savings already for comparatively small systems with just four modes.

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“…The fast approximation of tensors can be based on several decompositions of tensors such as: Tucker decomposition [43]; matricizations of tensors, as unfolding and applying SVD one time or several time recursively, (see below); higher order singular value decomposition (HOSVD) [3], Tensor-Train decompositions [35,36]; hierarchical Tucker decomposition [24,26]. A very recent survey [25] gives an overview on this dynamic field.…”

Section: Fast Approximation Of Tensorsmentioning

confidence: 99%

Low-Rank Approximation of Tensors

Friedland

Tammali

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained via the singular value decomposition, which allows to compute best rank k-approximations. For very big matrices a low rank approximation using SVD is not computationally feasible. In this case different approximations are available. It seems that variants of the CURdecomposition are most suitable.For d-mode tensors T ∈ ⊗ d i=1 R n i , with d > 2, many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. The most appropriate approximation seems to be best (r1, . . . , r d )-approximation, which maximizes the 2 norm of the projection of T on ⊗ d i=1 Ui, where Ui is an ri-dimensional subspace R n i . One of the most common methods is the alternating maximization method (AMM). It is obtained by maximizing on one subspace Ui, while keeping all other fixed, and alternating the procedure repeatedly for i = 1, . . . , d. Usually, AMM will converge to a local best approximation. This approximation is a fixed point of a corresponding map on Grassmannians. We suggest a Newton method for finding the corresponding fixed point. We also discuss variants of CUR-approximation method for tensors.The first part of the paper is a survey on low rank approximation of tensors. The second new part of this paper is a new Newton method for best (r1, . . . , r d )-approximation. We compare numerically different approximation methods.2000 Mathematics Subject Classification. 14M15, 15A18, 15A69, 65H10, 65K10.

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Hierarchical Singular Value Decomposition of Tensors

Cited by 469 publications

References 13 publications

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Multi-layer Potfit: An accurate potential representation for efficient high-dimensional quantum dynamics

Low-Rank Approximation of Tensors

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