AN ORDER-P TENSOR MULTIPLICATION WITH CIRCULANT STRUCTURE

ABSTRACT


INTRODUCTION
In the era of big data, multidimensional data analysis becomes more important, since much of realworld data is inherently multidimensional.Research on mathematical operations involving multidimensional arrays or tensors has increased along with the growing applications involving multidimensional data analysis.
The -product of order-3 tensor is one of the tensor multiplication operations defined by [1].The product is defined using two operations that transform the multiplication of two tensors into the multiplication of block matrices, then the result is a block matrix which is further transformed back into a tensor.Order-3 -product definition motivates further researchers to generalize its definition and define concepts on tensor over -product which analogous to concepts on matrix over standard matrix multiplication.Then, [1] defined identity, inverse, and transpose of order-3 tensor.Hereafter, [2] generalized the definition of -product, identity, inverse, and transpose of order-3 tensor into order- tensor (recursively) and constructed an algorithm to compute it using Fast Fourier Transform function in MATLAB.Later, [3] and [4] defined and discuss about Moore-Penrose inverse of tensor over -product and algorithm to compute it in MATLAB.Afterwards, some applications involving -product are discussed including image deblurring [2], [5], facial recognition [2], [6], and data compression [7], [8].Further and related discussion about -product discussed in articles [5], [6], [9]- [17].
A magnificent theorem formulated by [4] states that the order-3 -product between two tensors is fundamentally the multiplication of two block circulant matrices obtained by transforming the two tensors.The result of multiplication is then transformed back into a tensor using the inverse transformation of the same transformation.With this theorem, the concepts of tensors related to the -product can be viewed based on their circulant matrix forms.However, this theorem is still limited to order-3 tensors.
In this research, a more general theorem is proposed compared to the [4]'s theorem, so that it applies to tensors of arbitrary order or order- tensors.The proof is essentially a recursive version of proving [4]'s theorem, as the order- -product is a recursive version of the order-3 -product.The theorem provides a new and simpler perspective in understanding the definition of the order- -product and its related concepts and applications, which have been discussed by previously researchers.
In this research, several concepts of tensors over the -product will also be discussed based on their block circulant matrix forms, namely identity tensors, inverse, transpose, and Moore-Penrose inverse of tensor.Additionally, algorithms for calculating the -product involving the diagonalization of circulant matrices using the Discrete Fourier Transform (DFT) matrix will also be explored.

RESEARCH METHODS
In this research, several definitions and theorems that have been made by previous studies are used.In this section, these definitions and theorems will be given, which will be presented in several subsections.The first subsection will mention the basic definition and theorem of -product.The second subsection will present some definitions of concepts on the tensor over the -product operation.The last subsection will give a theorem that can be used to construct a MATLAB algorithm to calculate the -product.

Preliminary Definitions and Theorems
This section provides the definition of -product and the definition of some operations used on it, then followed by the theorem about the order-3 -product as the product of two circulant matrices.The theorem will be the basic material in this research.

Definition 1. [18]
An  ×  cyclic forward shift matrix, denoted by   is a matrix of the form Let  be a block matrix of the form with   of size  × , then block circulant matrix of  denoted by () and defined as follows .
where ⊗ is Kronecker product and   is an identity matrix of size .

Definition 3. [2]
The  operation works by taking a tensor  of size  1 ×  2 × . ..×   and converting it into a block tensor of size   × 1 with each block being a tensor of size  1 ×  2 × . ..×  −1 , as follows and  is the inverse operation of , which takes a block tensor of size   × 1 with each block being a tensor of size  1 ×  2 × . ..×  −1 , and then converts it into an  1 ×  2 × . ..×   tensor.Thus, we get
Finally, a theorem from [4] is given which will be generalized in this research.

Theorem 1. [4]
Suppose  is a tensor of size  1 ×  2 ×  3 and ℬ is a tensor of size  2 ×  ×  3 , the following holds By Theorem 1, the conceptualization of an order-3 -product can be construed as a multiplication operation involving block circulant.Using this theorem, the computation of the order-3 -product can be done through the utilization of a simpler structure, namely block circulant matrix.The generalization of Theorem 1 to encompass order- tensors represent a significant advancement since it will open up a new and simpler view of understanding the order- -product and its concepts and applications that have been discussed in previous researches.

Some concepts of tensor over 𝑡-product operation have been defined and discussed in previous researches [1]-[4], [9], [11]-[15], [17].
Here the definition of some concepts of tensor over -product that will be discussed more in this research.

𝒕-Product Algorithm
Some researches on -product algorithm have been done by [1]- [4], [15].In this research, we will also construct the -product algorithm to compute -product between two tensors and inverse Moore-Penrose of a tensor based on our proposed theorem.Here will be given some definitions and theorems of -product algorithm by previous researches.

RESULTS AND DISCUSSION
This research will be conducted in three subsections.The first subsection will create and prove a theorem that extends the theorem on the order-3 -product as a multiplication of the block circulant matrices form of the tensors to be multiplied by [4] (see Theorem 1), so that it applies to any order of tensor.The second subsection will discuss the concepts of tensor over -product based on its block circulant matrix form.The third subsection will discuss the algorithm for calculating the -product utilizing the diagonalization of the circulant matrix using the discrete Fourier transform matrix.

The Main Theorem
In this research, our main theorem will be recursive version of [4]'s with the main objective to calculate the order- -product as a standard matrix multiplication of block circulant matrices.As its recursive version, first, we will introduce some recursive operation notations that will help in proving the main theorem.
The  operation on , denoted () (using uppercase 'U'), works by applying the  operation on  repeatedly (recursively) until the tensor  turns into a matrix of  3  4 …   × 1 blocks with each block of size  1 ×  2 , and the result is denoted as  ̂.
The inverse operation of the  operation is  (using uppercase 'F'), which is an operation that works by performing the  operation repeatedly (recursively) on the block matrix the result of the tensor subjected to the  operation such that (()) = ( ̂) = .
• () =  ̃ The CU operation on  or () works by performing the  operation then  operation on  repeatedly (recursively) until the tensor  turns into a block matrix of size  3  4 …   ×  3  4 …   with each block of size  1 ×  2 , and the result is denoted by  ̃.

• 𝐹𝑈(𝐴 ̃) = 𝒜
The inverse operation of  is , which is an operation that works by performing the  and then  operation repeatedly (recursively) on the block matrix the result of the tensor subjected to the  operation such that (()) = ( ̃) = .
The following lemma is given before proving the main theorem.
Suppose  ⃗ is a vector, then based on the definition of circulant matrix, we obtain that the first column of ( ⃗) is  ⃗.The same is true for tensors, i.e., suppose  is a tensor, then the first column of (()) is (), so by a recursive process, it can be obtained that the first column of  ̃ is  ̂.For simplicity let  =  3  4 …   .Based on the definition of  operation, the second and onwards column of () is the result of multiplying the cyclic forward shift matrix with its the first column.This also applies to the block matrix  ̃, with the second and onwards columns of  ̃ being the result of multiplying the cyclic forward shift matrix with its first column which is  ̂.Based on Theorem 4, the order- -product  * ℬ can be calculated as By Equation ( 1), the -product is fundamentally a matrix multiplication between two block circulant matrices.The -product multiplication structure based on Theorem 4 can be more easily understood through the illustration below.This way to compute is simpler than using the definition, because it only uses one reversible transformation, that is the transformation from tensor domain into block circulant matrix domain.Theorem 1 also states that when we want to multiply two tensors, we must transform the tensor first into its block circulant matrix form.

Some Concepts of Tensor
As by Theorem 4 the -product of two tensor is essentially multiplication of block circulant matrix form of both tensors.In this subsection we will discuss more about some concepts of tensor namely identity, inverse, and transpose of tensor in the view of its block circulant matrix form.

• Identity tensor
By the Definition 6, if ℐ is an order- identity tensor of size  ×  ×  3 × ⋯ ×   , then the block circulant form of identity tensor is an identity matrix of size  3  4 …   .

𝐶𝑈(ℐ) = 𝐼 ̃= 𝐼 𝑛𝑛 3 𝑛 4 …𝑛 𝑝 .
By this and by Theorem 4, the -product between any tensor and the identity tensor is essentially the multiplication between any block circulant matrix and the identity matrix.This strengthens the results of previous research which proved that the identity tensor over the -product has properties analogous to the identity matrix over standard matrix multiplication.

• Inverse of tensor
By the Definition 7, if  −1 is inverse of tensor  then the block circulant matrix form of  −1 is inverse matrix of block circulant matrix form of .

𝐶𝑈(𝒜
By this and by Theorem 4, the -product between any tensor and its invers is essentially the multiplication between any block circulant matrix and its inverse.This strengthens the results of previous research which proved that the inverse tensor over the -product has properties analogous to the invers matrix over standard matrix multiplication.
• Transpose of tensor By the Definition 8, if   is transpose of tensor  then the block circulant matrix form of   is transpose matrix of block circulant matrix form of .
(  ) =   ̃=  ̃ = (())  By this and by Theorem 4, the -product between any tensor and its transpose is essentially the multiplication between any block circulant matrix and its invers.This strengthens the results of previous research which proved that the transpose tensor over the -product has properties analogous to the transpose matrix over standard matrix multiplication.
• Moore-Penrose inverse of tensor By the Definition 9, if  + is inverse Moore-Penrose of tensor  then the block circulant matrix form of  + is inverse Moore-Penrose of matrix of block circulant matrix form of .
( + ) =  + ̃=  ̃+ = (()) + By this and by Theorem 4, the -product between any tensor and its Moore-Penrose inverse is essentially the multiplication between any block circulant matrix and its Moore-Penrose inverse.This strengthens the results of previous research which proved that the Moore-Penrose inverse tensor over the -product has properties analogous to the Moore-Penrose inverse matrix over standard matrix multiplication.

𝒕-Product Algorithm
As Theorem 4 simplifies the equation for calculating the -product, Theorem 4 can also be used to simplify the equation for calculating -product in Theorem 3. The simpler form of the equation for calculating the -product in Theorem 3 will make it easier to understand the -product algorithm.This section will discuss the MATLAB algorithm for calculating the -product and the inverse Moore-Penrose of a tensor.The illustration to compute it can be seen in Figure 2. As on MATLAB, the block diagonal matrix results of diagonalization of a block circulant matrix that is ( ̃⊗   1 ) ̃( ̃ * ⊗   2 ) can be obtained by applying repeating FFTs along each mode of tensor .Using MATLAB one can compute -product by the following algorithm.

MATLAB algorithm to compute the 𝒕-product
Input:  1 ×  2 ×  3 × ⋯ ×   tensor  and  2 ×  ×  3 × ⋯ ×   tensor ℬ % Defining tensor  is done by writing each piece in the form of the matrix as (: , : ,  3 ,  4 , … ,   ), as well with ℬ SA = size(A); % stores the size of the tensor  as a vector SB = size(B); % stores the size of the tensor  as a vector

CONCLUSIONS
According the result and discussion above, there are three main points that can be concluded, which are as follows: 1.The -product of tensors fundamentally involves circulant matrix multiplication, which means that the operation at its core relies on multiplying circulant matrices.
2. The proposed theorem makes the concepts of tensor over -product more understandable to be analogues to the same concepts in matrix over standard multiplication.
3. The theorem simplifies the computation of -product and inverse Moore-Penrose of a tensor using matrix DFT.

Figure 1 .
Figure 1.Illustration to Compute -product Based On Proposed Theorem

Figure 2 .
Figure 2. Illustration to Compute -product using DFT Matrix