Third-Order Tensor Decorrelation Based on 3D FO-HKLT with Adaptive Directional Vectorization

In this work, we present a new hierarchical decomposition aimed at the decorrelation of a cubical tensor of size 2n, based on the 3D Frequency-Ordered Hierarchical KLT (3D FO-HKLT). The decomposition is executed in three consecutive stages. In the first stage, after adaptive directional vectorization (ADV) of the input tensor, the vectors are processed through one-dimensional FOAdaptive HKLT (FO-AHKLT), and after folding, the first intermediate tensor is calculated. In the second stage, on the vectors obtained after ADV of the first intermediate tensor, FO-AHKLT is applied, and after folding, the second intermediate tensor is calculated. In the third stage, on the vectors obtained from the second intermediate tensor, ADV is applied, followed by FO-AHKLT, and the output tensor is obtained. The orientation of the vectors, calculated from each tensor, could be horizontal, vertical or lateral. The best orientation is chosen through analysis of their covariance matrix, based on its symmetry properties. The kernel of FO-AHKLT is the optimal decorrelating KLT with a matrix of size 2 × 2. To achieve higher decorrelation of the decomposition components, the direction of the vectors obtained after unfolding of the input tensor in each of the three consecutive stages, is chosen adaptively. The achieved lower computational complexity of FO-AHKLT is compared with that of the Hierarchical Tucker and Tensor Train decompositions.


Introduction
The main tensor decompositions could be divided into two groups. The first group comprises decompositions executed in the spatial domain of the tensor. These are the famous Canonical Polyadic Decomposition (CPD), Higher-Order Singular Value Decomposition (HOSVD) [1][2][3][4], Tensor Train (TT) Decomposition [5], Hierarchical Tucker (H-Tucker) algorithm [6] and some of their modifications [7,8], based on the calculation of the tensor eigenvalues and eigenvectors. Their most important feature is that they are optimal regarding the minimization of the mean square approximation error derived from the low-energy component "truncation". The calculation of the retained components is based on iterative methods [9,10] that need a relatively small number of mathematical operations to achieve the requested accuracy. The hierarchical tensor decompositions based on the H-Tucker algorithm are presented in publications [8,11]. The compositional hierarchical tensor factorization introduced in [8] disentangles the hierarchical causal structure of object image formation, but the computational complexity (or Complexity) is not presented. In [11] is offered the TT-based hierarchical decomposition of high-order tensors, based on the Tensor-Train Hierarchical SVD (TT-HSVD). This approach permits parallel processing, which significantly accelerates the process. Unlike the TT-SVD algorithm, TT-HSVD is based on applying SVDs to matrices of smaller dimensions, which results in lower Complexity of TT-HSVD.
The second group comprises tensor decompositions performed in the transform domain, which use reversible 3D linear orthogonal transforms such as the Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinguished by its flexibility regarding the choice of the transform based on the processed data contents.
In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity. As a basis, we present here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through adaptive directional tensor vectorization (ADV).
In Section 2, we present the method for 3D hierarchical adaptive transform based on the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: Here, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The coefficients ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size as X . Each coefficient ) m,v,l ( s represents the weights of the basic tensor,  (1)) the main part of the tensor X power is concentrated, and their high decorrelation is achieved. The kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix of size 2 × 2. The decomposition comprises three consecutive stages arranged in accordance with the correlation analysis of the tensor X elements. One example decomposition for the tensor X of size N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT on the input tensor X, it is sequentially transformed into the intermediate tensors E, F and the output tensor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by using the FO-AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, the tensor X is transformed into the first intermediate tensor E, of the same size.
∈ R N×N×N is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: 2 of 15 comprises tensor decompositions performed in the transform doersible 3D linear orthogonal transforms such as the Fast Fourier crete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinity regarding the choice of the transform based on the processed data e present alternative new hierarchical 3D tensor decompositions s statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. mal (which ensures full decorrelation of the decomposition compoed iterations and have lower computational complexity. As a basis, ecomposition called 3D Frequency-Ordered Adaptive Hierarchical nsform (3D FO-AHKLT), whose efficiency is enhanced through ensor vectorization (ADV). present the method for 3D hierarchical adaptive transform based on Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Trans-Section 3 gives the details on the cubical tensor decomposition FO-AHKLT based on correlation analysis, and Section 4 explains . In Section 5, we analyze the computational complexity of the new d to that of the well-known H-Tucker and TT decompositions; Secnclusion. tensor X power is concentrated, and their high decorrelation is of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix ion comprises three consecutive stages arranged in accordance with is of the tensor X elements. One example decomposition for the ten-(for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT , it is sequentially transformed into the intermediate tensors E, F and The 3D FO-AHKLT is divisible, and this permits it to be executed by , whose graph for the case N = 8 is shown in Figure 2. As a result, sformed into the first intermediate tensor E, of the same size.
Here, N = 2 n is the size of the tensor The second group comprises tensor decompositions performed in the transform domain, which use reversible 3D linear orthogonal transforms such as the Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinguished by its flexibility regarding the choice of the transform based on the processed data contents.
In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity. As a basis, we present here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through adaptive directional tensor vectorization (ADV).
In Section 2, we present the method for 3D hierarchical adaptive transform based on the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: Here, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The coefficients ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size as X . Each coefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each basic tensor is represented as the outer product of three vectors: Here, "  " denotes the outer product of the two column-vectors are the basic vectors, obtained after execution of the three stages of 3D FO-HKLT. In the first decomposition components m,v,l l). v, s(m, K (as given in Equation (1)) the main part of the tensor X power is concentrated, and their high decorrelation is achieved. The kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix of size 2 × 2.
The decomposition comprises three consecutive stages arranged in accordance with the correlation analysis of the tensor X elements. One example decomposition for the tensor X of size N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT on the input tensor X, it is sequentially transformed into the intermediate tensors E, F and the output tensor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by using the FO-AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, the tensor X is transformed into the first intermediate tensor E, of the same size.
with nonnegative components x (i, j, k). The coefficients s(m, v, l) are the elements of the spectrum tensor S, which is of the same size as 2 of 15 or decompositions performed in the transform dor orthogonal transforms such as the Fast Fourier nsform (DCT), etc. [12][13][14]. This approach is distinchoice of the transform based on the processed data ative new hierarchical 3D tensor decompositions ogonal Karhunen-Loeve Transform (KLT) [15,16]. res full decorrelation of the decomposition compohave lower computational complexity. As a basis, alled 3D Frequency-Ordered Adaptive Hierarchical -AHKLT), whose efficiency is enhanced through ion (ADV). od for 3D hierarchical adaptive transform based on red Adaptive Hierarchical Karhunen-Loeve Transthe details on the cubical tensor decomposition sed on correlation analysis, and Section 4 explains e analyze the computational complexity of the new ell-known H-Tucker and TT decompositions; Secy-Ordered Hierarchical KLT of a Cubical Tensor position of 3rd order cubical tensor is Hierarchical Karhunen-Loeve Transform (3D FO- [17]: sor X with nonnegative components x (i, j, k). The s of the spectrum tensor S, which is of the same size sents the weights of the basic tensor, duct of the two column-vectors ree consecutive stages arranged in accordance with elements. One example decomposition for the tenown in Figure 1. After applying the 3D FO-AHKLT y transformed into the intermediate tensors E, F and LT is divisible, and this permits it to be executed by for the case N = 8 is shown in Figure 2. As a result, first intermediate tensor E, of the same size.
. Each coefficient s(m, v, l) represents the weights of the basic tensor, The second group comprises tensor decompositions performed in the transform domain, which use reversible 3D linear orthogonal transforms such as the Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinguished by its flexibility regarding the choice of the transform based on the processed data contents.
In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity. As a basis, we present here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through adaptive directional tensor vectorization (ADV).
In Section 2, we present the method for 3D hierarchical adaptive transform based on the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: Here, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The coefficients ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size as X . Each coefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each basic tensor is represented as the outer product of three vectors: Here, "  " denotes the outer product of the two column-vectors (1)) the main part of the tensor X power is concentrated, and their high decorrelation is achieved. The kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix of size 2 × 2. The decomposition comprises three consecutive stages arranged in accordance with the correlation analysis of the tensor X elements. One example decomposition for the tensor X of size N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT on the input tensor X, it is sequentially transformed into the intermediate tensors E, F and the output tensor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by using the FO-AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, the tensor X is transformed into the first intermediate tensor E, of the same size. u,v,l . Each basic tensor is represented as the outer product of three vectors:

of 15
second group comprises tensor decompositions performed in the transform doich use reversible 3D linear orthogonal transforms such as the Fast Fourier (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distiny its flexibility regarding the choice of the transform based on the processed data is work, we present alternative new hierarchical 3D tensor decompositions the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. close to optimal (which ensures full decorrelation of the decomposition compot do not need iterations and have lower computational complexity. As a basis, nt here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical -Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through directional tensor vectorization (ADV). ction 2, we present the method for 3D hierarchical adaptive transform based on imensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Trans--AHKLT). Section 3 gives the details on the cubical tensor decomposition eparable 3D FO-AHKLT based on correlation analysis, and Section 4 explains d algorithm. In Section 5, we analyze the computational complexity of the new es compared to that of the well-known H-Tucker and TT decompositions; Sectains the conclusion.

d for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
proposed method for decomposition of 3rd order cubical tensor is the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FOefined by the relation below [17]: , N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The ts ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size h coefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each or is represented as the outer product of three vectors: , "  " denotes the outer product of the two column-vectors part of the tensor X power is concentrated, and their high decorrelation is The kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix 2. decomposition comprises three consecutive stages arranged in accordance with ation analysis of the tensor X elements. One example decomposition for the tenize N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT ut tensor X, it is sequentially transformed into the intermediate tensors E, F and t tensor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by FO-AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, r X is transformed into the first intermediate tensor E, of the same size.
Here, "•" denotes the outer product of the two column-vectors (x • y = x·y T ), and k m (1), k v (2), k l (3) are the basic vectors, obtained after execution of the three stages of 3D FO-HKLT. In the first decomposition components s(m,v,l). The second group comprises tensor decompositions performed in the transform domain, which use reversible 3D linear orthogonal transforms such as the Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinguished by its flexibility regarding the choice of the transform based on the processed data contents.
In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity. As a basis, we present here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through adaptive directional tensor vectorization (ADV).
In Section 2, we present the method for 3D hierarchical adaptive transform based on the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: Here, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The coefficients ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size as X . Each coefficient ) m,v,l ( s represents the weights of the basic tensor, Each basic tensor is represented as the outer product of three vectors: Here, "  " denotes the outer product of the two column-vectors are the basic vectors, obtained after execution of the three stages of 3D FO-HKLT. In the first decomposition components m,v,l l). v, s(m, K (as given in Equation (1)) the main part of the tensor X power is concentrated, and their high decorrelation is achieved. The kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix of size 2 × 2.
The decomposition comprises three consecutive stages arranged in accordance with the correlation analysis of the tensor X elements. One example decomposition for the tensor X of size N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT on the input tensor X, it is sequentially transformed into the intermediate tensors E, F and the output tensor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by using the FO-AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, the tensor X is transformed into the first intermediate tensor E, of the same size. m,v,l (as given in Equation (1)) the main part of the tensor 2 of 15 second group comprises tensor decompositions performed in the transform dohich use reversible 3D linear orthogonal transforms such as the Fast Fourier rm (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinby its flexibility regarding the choice of the transform based on the processed data . this work, we present alternative new hierarchical 3D tensor decompositions n the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16].
close to optimal (which ensures full decorrelation of the decomposition compout do not need iterations and have lower computational complexity. As a basis, ent here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical en-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through e directional tensor vectorization (ADV). ection 2, we present the method for 3D hierarchical adaptive transform based on dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Trans-O-AHKLT). Section 3 gives the details on the cubical tensor decomposition separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains ed algorithm. In Section 5, we analyze the computational complexity of the new hes compared to that of the well-known H-Tucker and TT decompositions; Secntains the conclusion.

od for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
proposed method for decomposition of 3rd order cubical tensor is n the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FOdefined by the relation below [17]: re, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The nts ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size ach coefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each sor is represented as the outer product of three vectors: re, "  " denotes the outer product of the two column-vectors part of the tensor X power is concentrated, and their high decorrelation is . The kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix × 2. decomposition comprises three consecutive stages arranged in accordance with elation analysis of the tensor X elements. One example decomposition for the ten-power is concentrated, and their high decorrelation is achieved. The kernel of 3D FO-HKLT, defined as KLT 2×2 , is KLT with a transform matrix of size 2 × 2.
The decomposition comprises three consecutive stages arranged in accordance with the correlation analysis of the tensor The second group comprises tensor decompositions performed in the transform domain, which use reversible 3D linear orthogonal transforms such as the Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinguished by its flexibility regarding the choice of the transform based on the processed data contents.
In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity. As a basis, we present here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through adaptive directional tensor vectorization (ADV).
In Section 2, we present the method for 3D hierarchical adaptive transform based on the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: Here, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The coefficients ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size as X . Each coefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each basic tensor is represented as the outer product of three vectors: Here, "  " denotes the outer product of the two column-vectors are the basic vectors, obtained after execution of the three stages of 3D FO-HKLT. In the first decomposition components (as given in Equation (1)) the main part of the tensor X power is concentrated, and their high decorrelation is elements. One example decomposition for the tensor X of size N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT on the input tensor X, it is sequentially transformed into the intermediate tensors E, F and the output tensor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by using the FO-AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, the tensor 2 of 15 oup comprises tensor decompositions performed in the transform doreversible 3D linear orthogonal transforms such as the Fast Fourier iscrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinbility regarding the choice of the transform based on the processed data we present alternative new hierarchical 3D tensor decompositions ous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. ptimal (which ensures full decorrelation of the decomposition componeed iterations and have lower computational complexity. As a basis, e decomposition called 3D Frequency-Ordered Adaptive Hierarchical Transform (3D FO-AHKLT), whose efficiency is enhanced through al tensor vectorization (ADV). e present the method for 3D hierarchical adaptive transform based on al Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Trans-). Section 3 gives the details on the cubical tensor decomposition 3D FO-AHKLT based on correlation analysis, and Section 4 explains m. In Section 5, we analyze the computational complexity of the new red to that of the well-known H-Tucker and TT decompositions; Secconclusion.

Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
method for decomposition of 3rd order cubical tensor is requency-Ordered Hierarchical Karhunen-Loeve Transform (3D FOthe relation below [17]: s the size of the tensor X with nonnegative components x (i, j, k). The ) ,l are the elements of the spectrum tensor S, which is of the same size ient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each esented as the outer product of three vectors: , for q = 1, 2, 3, 4. For this case, the total The second group comprises tensor decompositions performed in the transform domain, which use reversible 3D linear orthogonal transforms such as the Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinguished by its flexibility regarding the choice of the transform based on the processed data contents.
In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity. As a basis, we present here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through adaptive directional tensor vectorization (ADV).
In Section 2, we present the method for 3D hierarchical adaptive transform based on the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: Here, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The coefficients ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size as X . Each coefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each basic tensor is represented as the outer product of three vectors: Here, "  " denotes the outer product of the two column-vectors  (1)) the main part of the tensor X power is concentrated, and their high decorrelation is achieved. The kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix of size 2 × 2. The decomposition comprises three consecutive stages arranged in accordance with the correlation analysis of the tensor X elements. One example decomposition for the tensor X of size N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT on the input tensor X, it is sequentially transformed into the intermediate tensors E, F and the output tensor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by using the FO-AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, the tensor X is transformed into the first intermediate tensor E, of the same size.
, for q = 1, 2, 3, 4. For this case, the tota The second group comprises tensor decompositions performed in the transform domain, which use reversible 3D linear orthogonal transforms such as the Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinguished by its flexibility regarding the choice of the transform based on the processed data contents.
In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity. As a basis, we present here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through adaptive directional tensor vectorization (ADV).
In Section 2, we present the method for 3D hierarchical adaptive transform based on the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: . The choice of the orientation direction of vectors x s (u) for s = 1, 2, . . . , N 2 is defined through analysis of their covariance . , e N,s ] T be the input and the output column-vectors of N = 2 n components, respectively, and n is the number of FO-HKLT hierarchical levels. The relation between vectors e s and x s , is [17]: where is the FO-HKLT matrix; P n (2 n ) of size 2 n × 2 n is the permutation matrix for the last level n of FO-HKLT, and n ∏ p=1 G p (2 n ) is the product of n sparse transform matrices G p (2 n ) for p = 1, 2, 3, . . . , n. Each matrix G p (2 n ) is defined as follows: for p = 1, 2, 3, . . . , n, where "⊕" denotes the direct sum of matrices.
In the level n, the components of the column-vectors y n,s = G n (2 n )y n−1,s (respectively, the components of matrices Y n,k for k = 0, 1, . . . , N − 1) are rearranged. For this, the permutation matrix P n (2 n ) is used.
From the components of the column-vectors e s = P n (2 n )y n,s are obtained the frequencyordered matrices E r (i.e., E 0 , E 1 , . . . , E N−1 ), calculated in accordance with the relation between their sequential number r and the sequential number k (for matrices Y n,k ).

Enhancement of the 3D FO-HKLT Efficiency, Based on Correlation Analysis
To achieve higher efficiency of the 3D tensor decomposition based on the 3D FO-HKLT, here, we use the correlation relations between the tensor The second group comprises tensor decompositions performed in the transform domain, which use reversible 3D linear orthogonal transforms such as the Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinguished by its flexibility regarding the choice of the transform based on the processed data contents.
In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity. As a basis, we present here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through adaptive directional tensor vectorization (ADV).
In Section 2, we present the method for 3D hierarchical adaptive transform based on the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: Here, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The components in the orthogonal directions x, y, z. To detect the direction in which these relations are strongest, correlation analysis should be used. It is based on the analysis of the covariance matrices K x (u) of vectors x s (u), obtained through unfolding mode u = 1, 2, 3 of the tensor the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: Here, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The coefficients ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size as X . Each coefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each basic tensor is represented as the outer product of three vectors: Here, "  " denotes the outer product of the two column-vectors  (1)) the main part of the tensor X power is concentrated, and their high decorrelation is achieved. The kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix of size 2 × 2.
The decomposition comprises three consecutive stages arranged in accordance with the correlation analysis of the tensor X elements. One example decomposition for the tensor X of size N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT on the input tensor X, it is sequentially transformed into the intermediate tensors E, F and the output tensor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by using the FO-AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, the tensor X is transformed into the first intermediate tensor E, of the same size.

Analysis of the Covariance Matrices
To estimate the correlation of vectors x s (u) = [x 1,s (u), x 2,s (u), . . . , x N,s (u)] T , for 3D FO-AHKLT, we use the ratio ∆(u) of the sums of the squares of coefficients k x,i,j (u) placed outside the main diagonal of the matrix K x (u), and those on the diagonal: The relation above takes into account the symmetry of coefficients k x,i,j (u)= k x,j,i (u) for i = j, in respect of the main diagonal of the covariance matrix, K x (u). The value of this ratio is maximum for the highest correlation of vectors x s (u) of same orientation, u.

Choice of Vectors' Orientation for Adaptive Directional Tensor Vectorization
The choice of the orientation u = 1, 2, 3 of vectors x s (u) in each stage S u of 3D FO-AHKLT which ensures maximum decorrelation for the tensor The second group comprises tensor decompositions performed in the transform domain, which use reversible 3D linear orthogonal transforms such as the Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinguished by its flexibility regarding the choice of the transform based on the processed data contents.
In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity. As a basis, we present here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through adaptive directional tensor vectorization (ADV).
In Section 2, we present the method for 3D hierarchical adaptive transform based on the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: Here, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The coefficients ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size as X . Each coefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each basic tensor is represented as the outer product of three vectors: Here, "  " denotes the outer product of the two column-vectors  (1)) the main part of the tensor X power is concentrated, and their high decorrelation is achieved. The kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix of size 2 × 2.
The decomposition comprises three consecutive stages arranged in accordance with the correlation analysis of the tensor X elements. One example decomposition for the tensor X of size N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT on the input tensor X, it is sequentially transformed into the intermediate tensors E, F and the output tensor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by using the FO-AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, the tensor X is transformed into the first intermediate tensor E, of the same size.
, is based on the relation between coefficients ∆(u), given in Table 1. The second group comprises tensor decompositions performed in the transform domain, which use reversible 3D linear orthogonal transforms such as the Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT), etc. [12][13][14]. This approach is distinguished by its flexibility regarding the choice of the transform based on the processed data contents.
In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity. As a basis, we present here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through adaptive directional tensor vectorization (ADV).
In Section 2, we present the method for 3D hierarchical adaptive transform based on the one-dimensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Transform (FO-AHKLT). Section 3 gives the details on the cubical tensor decomposition through separable 3D FO-AHKLT based on correlation analysis, and Section 4 explains the related algorithm. In Section 5, we analyze the computational complexity of the new approaches compared to that of the well-known H-Tucker and TT decompositions; Section 6 contains the conclusion.

Method for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
The proposed method for decomposition of 3rd order cubical tensor is based on the 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FO-HKLT), defined by the relation below [17]: Here, N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The coefficients ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size as X . Each coefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each basic tensor is represented as the outer product of three vectors: Here, "  " denotes the outer product of the two column-vectors  (1)) the main part of the tensor X power is concentrated, and their high decorrelation is achieved. The kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix of size 2 × 2.
The decomposition comprises three consecutive stages arranged in accordance with the correlation analysis of the tensor X elements. One example decomposition for the tensor X of size N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT on the input tensor X, it is sequentially transformed into the intermediate tensors E, F and the output tensor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by using the FO-AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, the tensor X is transformed into the first intermediate tensor E, of the same size.
, of size 4 × 4 × 4. The couples of elements of the vectors x q,s (u) for q = 1, 2 on which KLT 2×2 is applied in the lowest level of FO-HKLT, are colored in red and white. Table 1. Adaptive choice of the orientation of vectors x s (u) for u = 1, 2, 3, based on the correlation analysis.

Evaluation of the Decorrelation Properties of FO-AHKLT
The qualities of the one-dimensional FO-AHKLT correspond to the decorrelation degree of the output eigen images, which defines the number of needed execution levels, n. As an example, here is given the evaluation of the decorrelation properties of the transform, for the case N = 8 and n = 3. In the hierarchical level p = 1, 2, 3 of FO-AHKLT, the covariance matrix p y K of size 8 Figure 3. Orientations of vectors x s (u) for u = 1, 2, 3 and N = 4 after unfolding mode u = 1, 2, 3 of the tensor ing the choice of the transform based on the processed data alternative new hierarchical 3D tensor decompositions l orthogonal Karhunen-Loeve Transform (KLT) [15,16]. h ensures full decorrelation of the decomposition compons and have lower computational complexity. As a basis, tion called 3D Frequency-Ordered Adaptive Hierarchical D FO-AHKLT), whose efficiency is enhanced through torization (ADV). e method for 3D hierarchical adaptive transform based on -Ordered Adaptive Hierarchical Karhunen-Loeve Transgives the details on the cubical tensor decomposition LT based on correlation analysis, and Section 4 explains n 5, we analyze the computational complexity of the new f the well-known H-Tucker and TT decompositions; Secquency-Ordered Hierarchical KLT of a Cubical Tensor decomposition of 3rd order cubical tensor is dered Hierarchical Karhunen-Loeve Transform (3D FObelow [17]: he tensor X with nonnegative components x (i, j, k). The ements of the spectrum tensor S, which is of the same size represents the weights of the basic tensor, . u,v,l K Each e outer product of three vectors: (2) er product of the two column-vectors (as given in Equation X power is concentrated, and their high decorrelation is HKLT, defined as KLT2 × 2, is KLT with a transform matrix ises three consecutive stages arranged in accordance with nsor X elements. One example decomposition for the ten-8) is shown in Figure 1. After applying the 3D FO-AHKLT ntially transformed into the intermediate tensors E, F and -AHKLT is divisible, and this permits it to be executed by raph for the case N = 8 is shown in Figure 2. As a result, to the first intermediate tensor E, of the same size. to continue with the next level of FO-AHKLT, or to stop, must be taken. For this, the covariance matrix K p y (u) of the vectors y p,s (u) is analyzed for s = 1, 2, . . . , S, which defines the achieved decorrelation. In the case that the decorrelation is full, their matrix K p y (u) is diagonal, and the algorithm is stopped. The proposed adaptive control of FO-AHKLT permits the process to stop earlier, despite that full decorrelation is not achieved, if the result is satisfactory. The decision to stop the FO-AHKLT in the current level p is defined by the relation: Here, k y p ,i,j (u) is the (i, j)th element of the matrix K p y (u), and δ 1 is a threshold of small value, set in advance. In the case that the condition is satisfied, the calculations stop. Otherwise, the processing continues with the next FO-AHKLT level, p + 1. When the calculations for the second level are finished, the result is checked again, but in this case k y p+1 ,i,j (u) are the elements of the matrix K p+1 y (u) of the vectors y p+1,s (u), etc. Taking into account the condition (21), the FO-AHKLT matrix defined in Equation (3) turns into: . In these relations, δ 2 is a threshold of small value, set in advance. The condition (21), together with the adaptive KLT 2×2 performed in accordance with relations (23)-(26), permits reducing the number of calculations without worsening the 3D FO-AHKLT decorrelation properties.

Algorithm 3D FO-AHKLT
On the basis of the 3D FO-HKLT tensor decomposition, together with the adaptive control of the directional vectorization on the input, the intermediate and the output tensor, and the correlation analysis, the algorithm called 3D FO-AHKLT is developed as Algorithm 1:

Algorithm 1
Input: Third-order tensor on 2, we present the method for 3D hierarchical adaptive transform based on ensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Trans-HKLT). Section 3 gives the details on the cubical tensor decomposition arable 3D FO-AHKLT based on correlation analysis, and Section 4 explains lgorithm. In Section 5, we analyze the computational complexity of the new compared to that of the well-known H-Tucker and TT decompositions; Secns the conclusion.

r 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
posed method for decomposition of 3rd order cubical tensor is 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FOned by the relation below [17]: = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size oefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each is represented as the outer product of three vectors: (2)  " denotes the outer product of the two column-vectors of the tensor X power is concentrated, and their high decorrelation is e kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix omposition comprises three consecutive stages arranged in accordance with n analysis of the tensor X elements. One example decomposition for the ten-N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT tensor X, it is sequentially transformed into the intermediate tensors E, F and nsor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by -AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, is transformed into the first intermediate tensor E, of the same size.
of size N × N × N (N = 2 n ) with elements x(i, j, k), and threshold values, δ 1 , δ 2 ; The steps of the algorithm are given below: 1. Unfolding of the tensor ere the decomposition called 3D Frequency-Ordered Adaptive Hierarchical oeve Transform (3D FO-AHKLT), whose efficiency is enhanced through ectional tensor vectorization (ADV). on 2, we present the method for 3D hierarchical adaptive transform based on ensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Trans-HKLT). Section 3 gives the details on the cubical tensor decomposition arable 3D FO-AHKLT based on correlation analysis, and Section 4 explains lgorithm. In Section 5, we analyze the computational complexity of the new compared to that of the well-known H-Tucker and TT decompositions; Secns the conclusion. r 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor posed method for decomposition of 3rd order cubical tensor is e 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FOned by the relation below [17]: = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The X power is concentrated, and their high decorrelation is e kernel of 3D FO-HKLT, defined as KLT2 × 2, is KLT with a transform matrix omposition comprises three consecutive stages arranged in accordance with on analysis of the tensor X elements. One example decomposition for the ten-N × N × N (for N = 8) is shown in Figure 1. After applying the 3D FO-AHKLT tensor X, it is sequentially transformed into the intermediate tensors E, F and nsor, S. The 3D FO-AHKLT is divisible, and this permits it to be executed by -AHKLT, whose graph for the case N = 8 is shown in Figure 2. As a result, is transformed into the first intermediate tensor E, of the same size.  [12][13][14]. This approach is distinits flexibility regarding the choice of the transform based on the processed data s work, we present alternative new hierarchical 3D tensor decompositions the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. lose to optimal (which ensures full decorrelation of the decomposition compot do not need iterations and have lower computational complexity. As a basis, t here the decomposition called 3D Frequency-Ordered Adaptive Hierarchical -Loeve Transform (3D FO-AHKLT), whose efficiency is enhanced through irectional tensor vectorization (ADV). tion 2, we present the method for 3D hierarchical adaptive transform based on mensional Frequency-Ordered Adaptive Hierarchical Karhunen-Loeve Trans-AHKLT). Section 3 gives the details on the cubical tensor decomposition parable 3D FO-AHKLT based on correlation analysis, and Section 4 explains algorithm. In Section 5, we analyze the computational complexity of the new s compared to that of the well-known H-Tucker and TT decompositions; Secains the conclusion.

for 3D Adaptive Frequency-Ordered Hierarchical KLT of a Cubical Tensor
roposed method for decomposition of 3rd order cubical tensor is he 3D Frequency-Ordered Hierarchical Karhunen-Loeve Transform (3D FOfined by the relation below [17]: (1) N = 2 n is the size of the tensor X with nonnegative components x (i, j, k). The s ) m,v,l ( s are the elements of the spectrum tensor S, which is of the same size coefficient ) m,v,l ( s represents the weights of the basic tensor, . u,v,l K Each r is represented as the outer product of three vectors: "  " denotes the outer product of the two column-vectors As a result of the algorithm execution:

•
The main part of the tensor energy is concentrated into a small number of coefficients s(m, v, l) of the spectrum tensor S, for m, v, l = 0, 1, 2; • The decomposition components of the tensor are uncorrelated.

Comparative Evaluation of the Computational Complexity
For the computation of the covariance matrix K p y (u) = E{y p,s (u)·y T p,s (u)} − E{y p,s (u)}· E{y p,s (u)} T of size N × N for N = 2 n , in correspondence with Equation (17), A K p y (n) = 3 × 2 2n additions and M K p y (n) = 2 2n multiplications are needed. Then, the total number of the operations needed for the calculation of K p y (u) is: O K p y (n) = A K p y (n) + M K p y (n) = 4 × 2 2n for p = 1, 2, . . . , n For the n levels of the algorithm FO-HKLT is obtained: For the calculation of ∆ u (u) in correspondence with Equation (17), A ∆u (n) = 2 2n − 1 additions and M ∆ u (n) = 2 2n + 1 multiplications are needed. Then, O ∆ u (n) = A ∆ u (n) + M ∆ u (n) = 2 × 2 2n for u = 1, 2, 3 Hence, In accordance with [17], the computational complexity (or Complexity) of FO-HKLT is defined by taking into account the number of needed additions A FO HKLT (n) = 2 n−1 (5 × 2 2n + 1)n ≈ 2.5 × 2 3n n, and multiplications M FO HKLT (n) = 2 n−1 (7 × 2 2n + 5)n ≈ 3.5 × 2 3n n. Hence, the total number of operations needed for the 3D FO-HKLT execution is: The total number of operations needed for the execution of the algorithm 3D FO-AHKLT (without taking into consideration the possibility to stop the processing earlier than the last level) is:

O 3DFO
AHKLT (n) = A 3DFO AHKLT (n) + M 3DFO AHKLT (n) ≈ 2 2n+1 [2n(9 × 2 n−1 + 1) For the H-Tucker and TT decomposition of a cubical tensor of size N = 2 n , in accordance with [17], the number of needed operations is: O HT (n) = 2 3n (2 n+1 + 3), O TT (n) = 3 × 2 4n (36) Compared to H-Tucker and TT, the Complexity of the new decomposition decreases together with the growth of n. In the general case, the relative Complexity of 3D FO-AHKLT with respect to decompositions H-Tucker and TT is evaluated in accordance with the relations below: AHKLT (n) = 2 n−1 2 n+1 + 3 2n(9 × 2 n−1 + 1) + 3 (37) AHKLT (n) = 2 2n−1 3 2n(9 × 2 n−1 + 1) + 3 For example, for n = 8 the results are: ψ 1 (8) = 3.57 and ψ 2 (8) = 5.33. For the case when the new decomposition must have minimum Complexity, the transform kernel KLT 2×2 could be replaced by WHT of the same size (2 × 2), which in correspondence with Equation (10), is the particular case of KLT 2×2 for the fixed value of θ 1,2 = π/4. According to [19], the Complexity of the 3D Fast Walsh-Hadamard Transform The relative Complexity of 3D FO-AHKLT with respect to 3D-AFWHT is defined by the relation: = 2 2n(9 × 2 n−1 + 1) + 3 3 × 2 n n + 2(2n + 3) In Table 2 are given the values of coefficients ψ 1 (n), ψ 2 (n) and ψ 3 (n), calculated in accordance with Equations (36), (37) and (40), for n = 2, 3, . . . , 10. From the table, it is seen that for n > 5, the values of coefficients ψ 1 (n) and ψ 2 (n) are higher than "1" and grow fast together with n, while for the same values of n, the coefficient ψ 3 (n) increases a little in the range from 4 up to 6. From the comparison follow the conclusions below: • The new hierarchical decomposition has low Complexity, which decreases together with the growth of n faster than those of the H-Tucker and TT decompositions; • Significant reduction in the decomposition Complexity could be achieved through replacement of the kernel KLT 2×2 by WHT 2×2 . In this particular case, the decrease in the decomposition Complexity results in a lower decorrelation degree; • For the case n = 8, the Complexity of the algorithm 3D-AFWHT is approximately 6 times lower than that of the 3D FO-AHKLT; • The Complexity of algorithms 3D FO-AHKLT and 3D-AFWHT was evaluated without taking into consideration the use of the adaptive KLT 2×2 and the possibility to stop the execution prior to the maximum level n. Equations (35) and (40) give the maximum values of Complexity used for the evaluation of the compared algorithms.

Conclusions
This work presented the new algorithm 3D FO-AHKLT, aimed at the decorrelation of the elements of a cubical tensor of size N = 2 n . The Complexity of the algorithm was evaluated and compared with that of other, similar algorithms, and its efficiency was shown. The main qualities of the tensor decomposition 3D FO-AHKLT are: • Efficient decorrelation of the calculated components; • Concentration of the tensor energy into a small number of decomposition components; • Lack of iterations; • Low Complexity; • The capacity for parallel recursive implementation, which reduces the needed memory volume; • The capacity for additional significant Complexity reduction through the use of the algorithm 3D-AFWHT, depending on the needs of the implemented application.
The presented algorithms 3D FO-AHKLT and 3D-AFWHT could be generalized for tensors with three different dimensions 2 n 1 × 2 n 2 × 2 n 3 (i.e., for n 1 = n 2 = n 3 ). The choice of the offered hierarchical 3D decompositions depends on the requirements and limitations of their Complexity imposed by the application area.
The future investigations of 3D FO-AHKLT and 3D-AFWHT will be aimed at the evaluation of their characteristics compared to famous tensor decompositions, in order to define the best settings and to define the most efficient applications for tensor image compression, feature space reduction, filtration, analysis, search and recognition of multidimensional visual information, deep learning, etc. The future development of the presented algorithm will be aimed at applications related to tree tensor network [20], multiway array (tensor) data analysis [21,22], tensor decompositions in neural networks for tree-structured data, etc.