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Article

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

1
Faculty of Telecommunications, Department Radio Communications and Video Technologies, Technical University of Sofia, 1000 Sofia, Bulgaria
2
TK Engineering, 1582 Sofia, Bulgaria
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(10), 1605; https://doi.org/10.3390/sym12101605
Received: 6 August 2020 / Revised: 22 September 2020 / Accepted: 25 September 2020 / Published: 26 September 2020
(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)
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. View Full-Text
Keywords: cubical tensor decomposition; 3D hierarchical adaptive PCA transform; 3D Frequency-Ordered Hierarchical KLT; computational complexity cubical tensor decomposition; 3D hierarchical adaptive PCA transform; 3D Frequency-Ordered Hierarchical KLT; computational complexity
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MDPI and ACS Style

Kountchev, R.; Mironov, R.; Kountcheva, R. Complexity Estimation of Cubical Tensor Represented through 3D Frequency-Ordered Hierarchical KLT. Symmetry 2020, 12, 1605. https://doi.org/10.3390/sym12101605

AMA Style

Kountchev R, Mironov R, Kountcheva R. Complexity Estimation of Cubical Tensor Represented through 3D Frequency-Ordered Hierarchical KLT. Symmetry. 2020; 12(10):1605. https://doi.org/10.3390/sym12101605

Chicago/Turabian Style

Kountchev, Roumen, Rumen Mironov, and Roumiana Kountcheva. 2020. "Complexity Estimation of Cubical Tensor Represented through 3D Frequency-Ordered Hierarchical KLT" Symmetry 12, no. 10: 1605. https://doi.org/10.3390/sym12101605

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