Generating Polynomials and Symmetric Tensor Decompositions

Nie, Jiawang

doi:10.1007/s10208-015-9291-7

Cited by 59 publications

(63 citation statements)

References 23 publications

(49 reference statements)

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“…) This specific tensor is in

C^{3 \times 3 \times 3}

, corresponding to the following polynomial

\begin{matrix} bold-scriptT false(x^{4} false) & = 81 x_{0}^{4} + 17 x_{1}^{4} + 626 x_{2}^{4} - 144 x_{0} x_{1}^{2} x_{2} \\ + 216 x_{0}^{3} x_{1} - 108 x_{0}^{3} x_{2} \\ + 216 x_{0}^{2} x_{1}^{2} + 54 x_{0}^{2} x_{2}^{2} \\ + 96 x_{0} x_{1}^{3} - 12 x_{0} x_{2}^{3} - 52 x_{1}^{3} x_{2} \\ + 174 x_{1}^{2} x_{2}^{2} - 508 x_{1} x_{2}^{3} \\ + 72 x_{0} x_{1} x_{2}^{2} - 216 x_{0}^{2} x_{1} x_{2} . \end{matrix}

The dimension of the tensor is 3 whereas the size of its core tensor is 2. ( A tensor case studied in previous study. )This specific tensor is in

C^{5 \times 5 \times 5}

, with its components given by

T_{i_{1} i_{2} i_{3}} = i_{1} i_{2} i_{3} - i_{1} - i_{2} - i_{3} (0 ⩽ i_{1}, i_{2}, i_{3} ⩽ 4) <...>

…”

Section: Miscellaneous Discussion and Error Estimationsmentioning

confidence: 99%

“…)This specific tensor is in

C^{5 \times 5 \times 5}

, with its components given by

T_{i_{1} i_{2} i_{3}} = i_{1} i_{2} i_{3} - i_{1} - i_{2} - i_{3} (0 ⩽ i_{1}, i_{2}, i_{3} ⩽ 4) .

The dimension of the tensor is 5 whereas the size of its core tensor is 2. ( Another tensor case studied in previous study. )This specific tensor is in

C^{5 \times 5 \times 5 \times 5}

, with its components given by

T_{i_{1} i_{2} i_{3} i_{4}} = \tan (i_{1} i_{2} i_{3} i_{4}) (0 ⩽ i_{1}, i_{2}, i_{3}, i_{4} ⩽ 4) .

The dimension of the tensor is 5 whereas the size of the core tensor is 4.…”

Section: Miscellaneous Discussion and Error Estimationsmentioning

confidence: 99%

“…• (Another tensor case studied in previous study. 36 ) This specific tensor is in C 5×5×5×5 , with its components given by…”

Section: Miscellaneous Discussion and Error Estimationsmentioning

confidence: 99%

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Tensor and its tucker core: The invariance relationships

Jiang

Yang

Zhang

2017

Numerical Linear Algebra App

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In [13], Hillar and Lim famously demonstrated that "multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard". Despite many recent advancements, the state-of-the-art methods for computing such 'tensor analogues' still suffer severely from the curse of dimensionality. In this paper we show that the Tucker core of a tensor however, retains many properties of the original tensor, including the CP rank, the border rank, the tensor Schatten quasi norms, and the Z-eigenvalues. When the core tensor is smaller than the original tensor, this property leads to considerable computational advantages as confirmed by our numerical experiments. In our analysis, we in fact work with a generalized Tucker-like decomposition that can accommodate any full column-rank factor matrices.

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“…) This specific tensor is in

C^{3 \times 3 \times 3}

, corresponding to the following polynomial

\begin{matrix} bold-scriptT false(x^{4} false) & = 81 x_{0}^{4} + 17 x_{1}^{4} + 626 x_{2}^{4} - 144 x_{0} x_{1}^{2} x_{2} \\ + 216 x_{0}^{3} x_{1} - 108 x_{0}^{3} x_{2} \\ + 216 x_{0}^{2} x_{1}^{2} + 54 x_{0}^{2} x_{2}^{2} \\ + 96 x_{0} x_{1}^{3} - 12 x_{0} x_{2}^{3} - 52 x_{1}^{3} x_{2} \\ + 174 x_{1}^{2} x_{2}^{2} - 508 x_{1} x_{2}^{3} \\ + 72 x_{0} x_{1} x_{2}^{2} - 216 x_{0}^{2} x_{1} x_{2} . \end{matrix}

The dimension of the tensor is 3 whereas the size of its core tensor is 2. ( A tensor case studied in previous study. )This specific tensor is in

C^{5 \times 5 \times 5}

, with its components given by

T_{i_{1} i_{2} i_{3}} = i_{1} i_{2} i_{3} - i_{1} - i_{2} - i_{3} (0 ⩽ i_{1}, i_{2}, i_{3} ⩽ 4) <...>

…”

Section: Miscellaneous Discussion and Error Estimationsmentioning

confidence: 99%

“…)This specific tensor is in

C^{5 \times 5 \times 5}

, with its components given by

T_{i_{1} i_{2} i_{3}} = i_{1} i_{2} i_{3} - i_{1} - i_{2} - i_{3} (0 ⩽ i_{1}, i_{2}, i_{3} ⩽ 4) .

The dimension of the tensor is 5 whereas the size of its core tensor is 2. ( Another tensor case studied in previous study. )This specific tensor is in

C^{5 \times 5 \times 5 \times 5}

, with its components given by

T_{i_{1} i_{2} i_{3} i_{4}} = \tan (i_{1} i_{2} i_{3} i_{4}) (0 ⩽ i_{1}, i_{2}, i_{3}, i_{4} ⩽ 4) .

The dimension of the tensor is 5 whereas the size of the core tensor is 4.…”

Section: Miscellaneous Discussion and Error Estimationsmentioning

confidence: 99%

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Tensor and its tucker core: The invariance relationships

Jiang

Yang

Zhang

2017

Numerical Linear Algebra App

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show abstract

“…where F is indexed as in (2.3) and each p α is a coefficient. Let deg(p) denote the total degree of a polynomial p. As defined in [36], a polynomial g ∈ C[x] m is called a generating polynomial for F if…”

Section: Catalecticant Matricesmentioning

confidence: 99%

Low Rank Symmetric Tensor Approximations

Nie¹

2017

SIAM J. Matrix Anal. & Appl.

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Abstract. For a given symmetric tensor, we aim at finding a new one whose symmetric rank is small and that is close to the given one. There exist linear relations among the entries of low rank symmetric tensors. Such linear relations can be expressed by polynomials, which are called generating polynomials. We propose a new approach for computing low rank approximations by using generating polynomials. First, we estimate a set of generating polynomials that are approximately satisfied by the given tensor. Second, we find approximate common zeros of these polynomials. Third, we use these zeros to construct low rank tensor approximations. If the symmetric tensor to be approximated is sufficiently close to a low rank one, we show that the computed low rank approximations are quasi-optimal.

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“…Every symmetric tensor is a linear combination of rank-1 symmetric tensors [10]. We refer to [2,4,10,22,24] for symmetric tensor and [17,18] for general tensors.…”

Section: Introductionmentioning

confidence: 99%

Completely Positive Binary Tensors

2019

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A symmetric tensor is completely positive (CP) if it is a sum of tensor powers of nonnegative vectors. This paper characterizes completely positive binary tensors. We show that a binary tensor is completely positive if and only if it satisfies two linear matrix inequalities. This result can be used to determine whether a binary tensor is completely positive or not. When it is, we give an algorithm for computing its cp-rank and the decomposition. When the order is odd, we show that the cp-rank decomposition is unique. When the order is even, we completely characterize when the cp-rank decomposition is unique. We also discuss how to compute the nearest cp-approximation when a binary tensor is not completely positive.2010 Mathematics Subject Classification. Primary 15A69, 44A60, 90C22.

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Generating Polynomials and Symmetric Tensor Decompositions

Cited by 59 publications

References 23 publications

Tensor and its tucker core: The invariance relationships

Tensor and its tucker core: The invariance relationships

Low Rank Symmetric Tensor Approximations

Completely Positive Binary Tensors

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