A note on tensor chain approximation

Espig, Mike; Naraparaju, Kishore Kumar; Schneider, Jan

doi:10.1007/s00791-014-0218-7

Cited by 22 publications

(21 citation statements)

References 24 publications

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“…The partial trace operator can be used for describing the tensor chain (TC) decomposition [12,26] simply by slightly modifying the suggested TT representations. Properties of TC decomposition should be more investigated in the future work.…”

Section: Discussionmentioning

confidence: 99%

Fundamental tensor operations for large-scale data analysis using tensor network formats

Lee

Cichocki

2017

Multidim Syst Sign Process

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We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation techniques including Candecomp/Parafac (CP), Tucker, and tensor train (TT) decompositions with a number of mathematical and graphical representations. We also provide a brief review of mathematical properties of the TT decomposition as a low-rank approximation technique. With the aim of breaking the curseof-dimensionality in large-scale numerical analysis, we describe basic operations on large-scale vectors, matrices, and high-order tensors represented by TT decomposition. The proposed representations can be used for describing numerical methods based on TT decomposition for solving large-scale optimization problems such as systems of linear equations and symmetric eigenvalue problems.

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Section: Discussionmentioning

confidence: 99%

Fundamental tensor operations for large-scale data analysis using tensor network formats

Lee

Cichocki

2017

Multidim Syst Sign Process

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“…Algorithm 1 computes the quantity

δ : = \frac{ε ‖ T ‖_{F}}{\sqrt{d}}

and requires a manual input of a divisor r 0 of

{rank}_{δ} (T_{⟨ 1 ⟩})

. The choice of r 0 by Zhao et al (and also for a related algorithm based on the skeleton/cross approximation), is to minimize

| r_{0} - \frac{{rank}_{δ} (T_{⟨ 1 ⟩})}{r_{0}} |

, but examples (see Section 5.1) show that this can lead to suboptimal compression ratios.…”

Section: Conversion From Full Format To Tr‐formatmentioning

confidence: 99%

“…We convert a tensor given in full format into a TR‐representation and compute its storage cost. We compare the TT‐representation with r 0 =1, Algorithm 1 using a balanced representation with

r_{0} = argmin | r_{0} - \frac{{rank}_{δ} (T_{⟨ 1 ⟩})}{r_{0}} |

, and Algorithm 2, to Algorithm 3. We do not compare to other algorithms for TR‐decompositions, since these have already been compared to the TR‐SVD algorithm in the literature .…”

Section: Computational Experimentsmentioning

confidence: 99%

“… Importance of choice of TR‐rank and heuristic algorithm: We show that the compression ratio of the TR‐format is highly dependent (in our examples, up to more than one order of magnitude) on the choice of the rank r 0 in Equation . In earlier work, this choice was always kept fixed and not adapted to the underlying tensor, but finding the optimal r 0 is crucial for the efficiency of the TR‐format. Using the notion of minimal TR‐ranks defined below, we clarify why this is the case.…”

Section: Introductionmentioning

confidence: 99%

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On algorithms for and computing with the tensor ring decomposition

Mickelin

Karaman

2020

Numerical Linear Algebra App

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Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high-dimensional data, achieving linear scaling with the input dimension instead of exponential scaling. In this paper, we investigate even lower storage-cost representations in the tensor ring format, which is an extension of the tensor train format with variable end-ranks. Firstly, we introduce two algorithms for converting a tensor in full format to tensor ring format with low storage cost. Secondly, we detail a rounding operation for tensor rings and show how this requires new definitions of common linear algebra operations in the format to obtain storage-cost savings. Lastly, we introduce algorithms for transforming the graph structure of graph-based tensor formats, with orders of magnitude lower complexity than existing literature. The efficiency of all algorithms is demonstrated on a number of numerical examples, and in certain cases, we demonstrate significantly higher compression ratios when compared to previous approaches to using the tensor ring format. K E Y W O R D Sgraph-based tensor formats, tensor format conversions, tensor-ring format, tensors

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“…The canonical polyadic decomposition (CPD) [2,15,16] and the Tucker decomposition [2,27] both generalize the notion of the matrix singular value decomposition (SVD) to higher-order tensors and have, therefore, received a lot of attention. More recent tensor decompositions are the tensor train (TT) [8,9,18,21] and the hierarchical Tucker decomposition [12,13]. It turns out that the latter two decompositions were already known in the quantum mechanics and condensed matter physics communities as the matrix product state (MPS) [23] and the tensor tree network (TTN) [25], respectively.…”

Section: Introductionmentioning

confidence: 99%

MERACLE: Constructive Layer-Wise Conversion of a Tensor Train into a MERA

Batselier

Cichocki

Wong

2020

Commun. Appl. Math. Comput.

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In this article, two new algorithms are presented that convert a given data tensor train into either a Tucker decomposition with orthogonal matrix factors or a multi-scale entanglement renormalization ansatz (MERA). The Tucker core tensor is never explicitly computed but stored as a tensor train instead, resulting in both computationally and storage efficient algorithms. Both the multilinear Tucker-ranks as well as the MERA-ranks are automatically determined by the algorithm for a given upper bound on the relative approximation error. In addition, an iterative algorithm with low computational complexity based on solving an orthogonal Procrustes problem is proposed for the first time to retrieve optimal rank-lowering disentangler tensors, which are a crucial component in the construction of a low-rank MERA. Numerical experiments demonstrate the effectiveness of the proposed algorithms together with the potential storage benefit of a low-rank MERA over a tensor train.

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A note on tensor chain approximation

Cited by 22 publications

References 24 publications

Fundamental tensor operations for large-scale data analysis using tensor network formats

Fundamental tensor operations for large-scale data analysis using tensor network formats

On algorithms for and computing with the tensor ring decomposition

MERACLE: Constructive Layer-Wise Conversion of a Tensor Train into a MERA

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