Symmetry

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13 pages, 353 KiB

Open AccessArticle

Examples on the Non-Uniqueness of the Rank 1 Tensor Decomposition of Rank 4 Tensors

by Edoardo Ballico

Symmetry 2022, 14(9), 1889; https://doi.org/10.3390/sym14091889 - 9 Sep 2022

Cited by 1 | Viewed by 1603

Abstract

We discuss the non-uniqueness of the rank 1 tensor decomposition for rank 4 tensors of format

m_{1} \times \dots \times m_{k}

,

k \geq 3

. We discuss several classes of examples and provide a complete classification if

m_{1} = m_{2} = 4

. Full article

(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)

23 pages, 799 KiB

Open AccessArticle

Analysis of Hypergraph Signals via High-Order Total Variation

by Ruyuan Qu, Hui Feng, Chongbin Xu and Bo Hu

Symmetry 2022, 14(3), 543; https://doi.org/10.3390/sym14030543 - 7 Mar 2022

Cited by 1 | Viewed by 2989

Abstract

Beyond pairwise relationships, interactions among groups of agents do exist in many real-world applications, but they are difficult to capture by conventional graph models. Generalized from graphs, hypergraphs have been introduced to describe such high-order group interactions. Inspired by graph signal processing (GSP) [...] Read more.

Beyond pairwise relationships, interactions among groups of agents do exist in many real-world applications, but they are difficult to capture by conventional graph models. Generalized from graphs, hypergraphs have been introduced to describe such high-order group interactions. Inspired by graph signal processing (GSP) theory, an existing hypergraph signal processing (HGSP) method presented a spectral analysis framework relying on the orthogonal CP decomposition of adjacency tensors. However, such decomposition may not exist even for supersymmetric tensors. In this paper, we propose a high-order total variation (HOTV) form of a hypergraph signal (HGS) as its smoothness measure, which is a hyperedge-wise measure aggregating all signal values in each hyperedge instead of a pairwise one in most existing work. Further, we propose an HGS analysis framework based on the Tucker decomposition of the hypergraph Laplacian induced by the aforementioned HOTV. We construct an orthonormal basis from the HOTV, by which a new spectral transformation of the HGS is introduced. Then, we design hypergraph filters in both vertex and spectral domains correspondingly. Finally, we illustrate the advantages of the proposed framework by applications in label learning. Full article

(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)

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19 pages, 906 KiB

Open AccessArticle

Incremental Nonnegative Tucker Decomposition with Block-Coordinate Descent and Recursive Approaches

by Rafał Zdunek and Krzysztof Fonał

Symmetry 2022, 14(1), 113; https://doi.org/10.3390/sym14010113 - 9 Jan 2022

Cited by 8 | Viewed by 2306

Abstract

Nonnegative Tucker decomposition (NTD) is a robust method used for nonnegative multilinear feature extraction from nonnegative multi-way arrays. The standard version of NTD assumes that all of the observed data are accessible for batch processing. However, the data in many real-world applications are [...] Read more.

Nonnegative Tucker decomposition (NTD) is a robust method used for nonnegative multilinear feature extraction from nonnegative multi-way arrays. The standard version of NTD assumes that all of the observed data are accessible for batch processing. However, the data in many real-world applications are not static or are represented by a large number of multi-way samples that cannot be processing in one batch. To tackle this problem, a dynamic approach to NTD can be explored. In this study, we extend the standard model of NTD to an incremental or online version, assuming volatility of observed multi-way data along one mode. We propose two computational approaches for updating the factors in the incremental model: one is based on the recursive update model, and the other uses the concept of the block Kaczmarz method that belongs to coordinate descent methods. The experimental results performed on various datasets and streaming data demonstrate high efficiently of both algorithmic approaches, with respect to the baseline NTD methods. Full article

(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)

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16 pages, 361 KiB

Open AccessArticle

Base Point Freeness, Uniqueness of Decompositions and Double Points for Veronese and Segre Varieties

by Edoardo Ballico

Symmetry 2021, 13(12), 2344; https://doi.org/10.3390/sym13122344 - 6 Dec 2021

Viewed by 2122

Abstract

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and [...] Read more.

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and tangent vectors of the Segre variety associated with the format of the tensor. We give complete results for unions of one point and one tangent vector. Full article

(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)

27 pages, 2549 KiB

Open AccessEditor’s ChoiceArticle

Tensor-Based Adaptive Filtering Algorithms

by Laura-Maria Dogariu, Cristian-Lucian Stanciu, Camelia Elisei-Iliescu, Constantin Paleologu, Jacob Benesty and Silviu Ciochină

Symmetry 2021, 13(3), 481; https://doi.org/10.3390/sym13030481 - 15 Mar 2021

Cited by 27 | Viewed by 3967

Abstract

Tensor-based signal processing methods are usually employed when dealing with multidimensional data and/or systems with a large parameter space. In this paper, we present a family of tensor-based adaptive filtering algorithms, which are suitable for high-dimension system identification problems. The basic idea is [...] Read more.

Tensor-based signal processing methods are usually employed when dealing with multidimensional data and/or systems with a large parameter space. In this paper, we present a family of tensor-based adaptive filtering algorithms, which are suitable for high-dimension system identification problems. The basic idea is to exploit a decomposition-based approach, such that the global impulse response of the system can be estimated using a combination of shorter adaptive filters. The algorithms are mainly tailored for multiple-input/single-output system identification problems, where the input data and the channels can be grouped in the form of rank-1 tensors. Nevertheless, the approach could be further extended for single-input/single-output system identification scenarios, where the impulse responses (of more general forms) can be modeled as higher-rank tensors. As compared to the conventional adaptive filters, which involve a single (usually long) filter for the estimation of the global impulse response, the tensor-based algorithms achieve faster convergence rate and tracking, while also providing better accuracy of the solution. Simulation results support the theoretical findings and indicate the advantages of the tensor-based algorithms over the conventional ones, in terms of the main performance criteria. Full article

(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)

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16 pages, 854 KiB

Open AccessArticle

Complexity Estimation of Cubical Tensor Represented through 3D Frequency-Ordered Hierarchical KLT

by Roumen Kountchev, Rumen Mironov and Roumiana Kountcheva

Symmetry 2020, 12(10), 1605; https://doi.org/10.3390/sym12101605 - 26 Sep 2020

Cited by 3 | Viewed by 2430

Abstract

In this work is introduced one new hierarchical decomposition for cubical tensor of size 2ⁿ, based on the well-known orthogonal transforms Principal Component Analysis and Karhunen–Loeve Transform. The decomposition is called 3D Frequency-Ordered Hierarchical KLT (3D-FOHKLT). It is separable, and its [...] Read more.

In this work is introduced one new hierarchical decomposition for cubical tensor of size 2ⁿ, based on the well-known orthogonal transforms Principal Component Analysis and Karhunen–Loeve Transform. The decomposition is called 3D Frequency-Ordered Hierarchical KLT (3D-FOHKLT). It is separable, and its calculation is based on the one-dimensional Frequency-Ordered Hierarchical KLT (1D-FOHKLT) applied on a sequence of matrices. The transform matrix is the product of n sparse matrices, symmetrical at the point of their main diagonal. In particular, for the case in which the angles which define the transform coefficients for the couples of matrices in each hierarchical level of 1D-FOHKLT are equal to π/4, the transform coincides with this of the frequency-ordered 1D Walsh–Hadamard. Compared to the hierarchical decompositions of Tucker (H-Tucker) and the Tensor-Train (TT), the offered approach does not ensure full decorrelation between its components, but is close to the maximum. On the other hand, the evaluation of the computational complexity (CC) of the new decomposition proves that it is lower than that of the above-mentioned similar approaches. In correspondence with the comparison results for H-Tucker and TT, the CC decreases fast together with the increase of the hierarchical levels’ number, n. An additional advantage of 3D-FOHKLT is that it is based on the use of operations of low complexity, while the similar famous decompositions need large numbers of iterations to achieve the coveted accuracy. Full article

(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)

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12 pages, 1594 KiB

Open AccessArticle

An Accelerated Symmetric Nonnegative Matrix Factorization Algorithm Using Extrapolation

by Peitao Wang, Zhaoshui He, Jun Lu, Beihai Tan, YuLei Bai, Ji Tan, Taiheng Liu and Zhijie Lin

Symmetry 2020, 12(7), 1187; https://doi.org/10.3390/sym12071187 - 17 Jul 2020

Cited by 2 | Viewed by 2533

Abstract

Symmetric nonnegative matrix factorization (SNMF) approximates a symmetric nonnegative matrix by the product of a nonnegative low-rank matrix and its transpose. SNMF has been successfully used in many real-world applications such as clustering. In this paper, we propose an accelerated variant of the [...] Read more.

Symmetric nonnegative matrix factorization (SNMF) approximates a symmetric nonnegative matrix by the product of a nonnegative low-rank matrix and its transpose. SNMF has been successfully used in many real-world applications such as clustering. In this paper, we propose an accelerated variant of the multiplicative update (MU) algorithm of He et al. designed to solve the SNMF problem. The accelerated algorithm is derived by using the extrapolation scheme of Nesterov and a restart strategy. The extrapolation scheme plays a leading role in accelerating the MU algorithm of He et al. and the restart strategy ensures that the objective function of SNMF is monotonically decreasing. We apply the accelerated algorithm to clustering problems and symmetric nonnegative tensor factorization (SNTF). The experiment results on both synthetic and real-world data show that it is more than four times faster than the MU algorithm of He et al. and performs favorably compared to recent state-of-the-art algorithms. Full article

(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)

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17 pages, 1909 KiB

Open AccessArticle

Hierarchical Cubical Tensor Decomposition through Low Complexity Orthogonal Transforms

by Roumen K. Kountchev, Rumen P. Mironov and Roumiana A. Kountcheva

Symmetry 2020, 12(5), 864; https://doi.org/10.3390/sym12050864 - 25 May 2020

Cited by 7 | Viewed by 4455

Abstract

In this work, new approaches are proposed for the 3D decomposition of a cubical tensor of size N × N × N for N = 2ⁿ through hierarchical deterministic orthogonal transforms with low computational complexity, whose kernels are based on the Walsh-Hadamard [...] Read more.

In this work, new approaches are proposed for the 3D decomposition of a cubical tensor of size N × N × N for N = 2ⁿ through hierarchical deterministic orthogonal transforms with low computational complexity, whose kernels are based on the Walsh-Hadamard Transform (WHT) and the Complex Hadamard Transform (CHT). On the basis of the symmetrical properties of the real and complex Walsh-Hadamard matrices are developed fast computational algorithms whose computational complexity is compared with that of the famous deterministic transforms: the 3D Fast Fourier Transform, the 3D Discrete Wavelet Transform and the statistical Hierarchical Tucker decomposition. The comparison results show the lower computational complexity of the offered algorithms. Additionally, they ensure the high energy concentration of the original tensor into a small number of coefficients of the so calculated transformed spectrum tensor. The main advantage of the proposed algorithms is the reduction of the needed calculations due to the low number of hierarchical levels compared to the significant number of iterations needed to achieve the required decomposition accuracy based on the statistical methods. The choice of the 3D hierarchical decomposition is defined by the requirements and limitations related to the corresponding application area. Full article

(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)

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7 pages, 239 KiB

Open AccessFeature PaperArticle

Non-Degeneracy of 2-Forms and Pfaffian

by Jae-Hyouk Lee

Symmetry 2020, 12(2), 280; https://doi.org/10.3390/sym12020280 - 13 Feb 2020

Viewed by 2468

Abstract

In this article, we study the non-degeneracy of 2-forms (skew symmetric

(0, 2)

-tensor)

α

along the Pfaffian of

α

. We consider a symplectic vector space

V

with a non-degenerate skew symmetric

(0, 2)

-tensor [...] Read more.

In this article, we study the non-degeneracy of 2-forms (skew symmetric

(0, 2)

-tensor)

α

along the Pfaffian of

α

. We consider a symplectic vector space

V

with a non-degenerate skew symmetric

(0, 2)

-tensor

ω

, and derive various properties of the Pfaffian of

α

. As an application we show the non-degenerate skew symmetric

(0, 2)

-tensor

ω

has a property of rigidity that it is determined by its exterior power. Full article

(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)

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