On the Largest Multilinear Singular Values of Higher-Order Tensors

Domanov, Ignat; Stegeman, Alwin; Lathauwer, Lieven De

doi:10.1137/16m110770x

Cited by 2 publications

(5 citation statements)

References 19 publications

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“…For tensors, this is not straightforward. It is of fundamental importance to understand the behavior of multilinear singular values . In this section, we show how one can construct an all‐orthogonal tensor

scriptT \in R^{I \times J \times K}

with particular multilinear singular values using an LS‐CPD.…”

Section: Applicationsmentioning

confidence: 99%

Linear systems with a canonical polyadic decomposition constrained solution: Algorithms and applications

Bousse

Vervliet

Domanov

et al. 2018

Numerical Linear Algebra App

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Summary Real‐life data often exhibit some structure and/or sparsity, allowing one to use parsimonious models for compact representation and approximation. When considering matrix and tensor data, low‐rank models such as the (multilinear) singular value decomposition, canonical polyadic decomposition (CPD), tensor train, and hierarchical Tucker model are very common. The solution of (large‐scale) linear systems is often structured in a similar way, allowing one to use compact matrix and tensor models as well. In this paper, we focus on linear systems with a CPD‐constrained solution (LS‐CPD). Our main contribution is the development of optimization‐based and algebraic methods to solve LS‐CPDs. Furthermore, we propose a condition that guarantees generic uniqueness of the obtained solution. We also show that LS‐CPDs provide a broad framework for the analysis of multilinear systems of equations. The latter are a higher‐order generalization of linear systems, similar to tensor decompositions being a generalization of matrix decompositions. The wide applicability of LS‐CPDs in domains such as classification, multilinear algebra, and signal processing is illustrated.

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scriptT \in R^{I \times J \times K}

with particular multilinear singular values using an LS‐CPD.…”

Section: Applicationsmentioning

confidence: 99%

Linear systems with a canonical polyadic decomposition constrained solution: Algorithms and applications

Bousse

Vervliet

Domanov

et al. 2018

Numerical Linear Algebra App

View full text Add to dashboard Cite

show abstract

“…In our tensor train case, which to the best of our knowledge has not been dealt with before, honeycombs as well as [4] fortunately provide both a theoretical and practical resolution to the simpler pairwise problem (see Section 1.3). The connection of feasibility to the Horn conjecture has, to a smaller extent, also synchronously and again independently been investigated by the afore mentioned article [6], as they deal with yet different eigenvalue problems. As already mentioned in Section 1.2, an analogous way of decoupling can be applied to the Tucker case and indeed any other hierarchical format, so that any such feasibility problem for a dth-order tensor can be reduced to the pairwise problems as in the tensor train format and/or the Tucker format in three dimensions.…”

Section: Other Results On the Feasibility Problemmentioning

confidence: 93%

“…However due to the difference between the two mentioned formats, no results could, so far, be transferred. Through matrix analysis and eigenvalue relations, [6] later introduced necessary and sufficient linear inequalities regarding feasibility mostly restricted to the largest Tucker-singular values of tensors with one common mode size. Independently, [30] proved the same result for the Tucker format provided n 1 = .…”

Section: Other Results On the Feasibility Problemmentioning

confidence: 99%

“…is the same as the Tucker-feasibility problem (cf. [6]). Since the tensor space is effectively reduced by one dimension, one may substitute d ← d − 1 and use K = {{1}, .…”

Section: Equivalence Of the Tensor Feasibility And The Quantum Margin...mentioning

confidence: 99%

“…This definition is straight forwardly extended to more than two summands. 6 The relation Eq. (3.2) may equivalently be written as λ µ (−ν) ∼ c 0 (cf.…”

Section: Weyl's Problem and The Horn Conjecturementioning

confidence: 99%

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A Geometric Description of Feasible Singular Values in the Tensor Train Format

Krämer¹

2019

SIAM J. Matrix Anal. Appl.

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Tree tensor networks such as the tensor train format are a common tool for high dimensional problems. The associated multivariate rank and accordant tuples of singular values are based on different matricizations of the same tensor. While the behavior of such is as essential as in the matrix case, here the question about the feasibility of specific constellations arises: which prescribed tuples can be realized as singular values of a tensor and what is this feasible set? We first show the equivalence of the tensor feasibility problem (TFP) to the quantum marginal problem (QMP). In higher dimensions, in case of the tensor train (TT-)format, the conditions for feasibility can be decoupled. By present results for three dimensions for the QMP, it then follows that the tuples of squared, feasible TT-singular values form polyhedral cones. We further establish a connection to eigenvalue relations of sums of Hermitian matrices, which in turn are described by sets of interlinked, so called honeycombs, as they have been introduced by Knutson and Tao. Besides a large class of universal, necessary inequalities as well as the vertex description for a special, simpler instance, we present a linear programming algorithm to check feasibility and a simple, heuristic algorithm to construct representations of tensors with prescribed, feasible TT-singular values in parallel.

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On the Largest Multilinear Singular Values of Higher-Order Tensors

Abstract: Let σn denote the largest mode-n multilinear singular value of an I 1 × · · · × I N tensor T . We prove that * Submitted to the editors DATE.

Cited by 2 publications

References 19 publications

Linear systems with a canonical polyadic decomposition constrained solution: Algorithms and applications

Linear systems with a canonical polyadic decomposition constrained solution: Algorithms and applications

A Geometric Description of Feasible Singular Values in the Tensor Train Format

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