Decomposition theorem of fuzzy tensors and its applications

Abstract

In this paper, we focus on investigating decomposition theorem of fuzzy tensors. We first propose definitions of cutting tensor and anti-fuzzy cutting tensor. Scalar product and anti-fuzzy scalar product of number and fuzzy tensor are put forth, and some properties of scalar product are discussed. Then, we propose and prove decomposition theorem of fuzzy tensor. Our numerical examples and applications show that the feasibility of the presented decomposition methods.

Keywords

Fuzzy tensor cutting tensor anti-fuzzy cutting tensor fuzzy tensor decomposition theorem MSC: 03E72 15B15 80M25

1 Introduction

Since 2005, tensor theory has gained more and more attention in the field of applied mathematics (see [1, 2]). Many experts and scholars have made significant progresses on tensor decompositions and applications. The research of tensor decompositions originated from the work of Hitchcock [3, 4] in 1927, since then there have developed many works about tensor decompositions [5 –14], which gradually improve the tensor decompositions theory. Tensor decompositions are higher-order analogues of matrix decompositions, and have been considered as useful tools for data analysis. Recent progresses on graph analysis are playing an increasingly important role in science and industry research. Most of the current studies of tensor decompositions include Tucker decompositions, CP decompositions and TT(Tensor-Train) decompositions. In particular, Tucker decomposition is a high-order form of principle component analysis. The Tucker decompositions [5, 6] is an approximate decomposition in the form of a core tensor multiplied by a different matrix along each direction. However, it is not easy to be calculated for higher order tensor. The CP decomposition [7, 9] is an approximate decomposition that is the sum of finite number of rank 1 tensor. The decomposition form is simple, but it is difficult to calculate the rank of the tensor. The TT decompositions [12, 13] are those in which the m order tensor is decomposed into the form of the product of m third order core tensor. It breaks the requirements of the order of Tucker decomposition, but needs more computation.

The pioneering work on fuzzy sets was provided by Zadeh [15] in 1965. The fuzzy set theory has been widely applied in various fields of modern society [16, 17]. Cut sets theory plays an important role in the research of fuzzy sets [18 –22], and systematic study of decomposition theorem of fuzzy sets can be found in literature [19, 22]. Especially, Fan and Liu [20, 21] have established decomposition theorem of fuzzy matrix, however, cut sets theory is still a primary ingredient to analyze fuzzy matrix. According to some practical applications, anti-fuzzy theory [23, 24] has become a new theoretical foundation for solving the problem of algebraic theory and information computing. Chen and Lu [25, 26] have described a new tensor – fuzzy tensor which is an upgraded of fuzzy matrix with respect to order, and a generalization of classic tensor. The Power Method and the Minimal Strong Component that two algorithms for solving the period and index of fuzzy tensor are obtained in literature [25], Order Reduction and Element Method that two algorithms for identifying the convergence of fuzzy tensors are presented in [26].

In this paper, We first use cutting tensor and scalar product to decompose the fuzzy tensor, and obtain decomposition theorem I. Besides, we apply anti-fuzzy ideas to figure out decomposition of fuzzy tensor, and obtain decomposition theorem II. We can accelerate decomposition by taking advantage of cutting tensor and anti-cutting tensor of fuzzy tensor. In the next section, we shall give the definitions of cutting tensor and anti-cutting tensor and some properties of scalar product and anti-fuzzy scalar product which are used for decomposition theorem. In section 3, decomposition theorem I is proposed and proved. One main and simple decomposition algorithm is presented. A numerical example show that the decomposition algorithm is accurate. And decomposition theorem II is introduced based on anti-cutting tensor and anti-fuzzy scalar product. In section 4, real applications of Congou black tea are expanded. In the last section, some conclusions will be reported.

2 Fuzzy tensor and cutting tensor

In this section, we introduce the definition of fuzzy tensor, and put forward the definitions of cutting tensor and anti-cutting tensor of fuzzy tensor. Moreover, we analyse some properties of cutting tensor which will be used in this paper.

As was in [1], a real order m dimension n tensor $A = (a_{i_{1} \dots i_{m}})$ consists of n^m entries: a_{i
₁} ⋯ _{i
_m} ∈ R,

where 1 ≤ i_j ≤ n for j = 1, ⋯ , m. It is obvious that a matrix is an order 2 tensor. Following the definition of the tensor, we give the definition of fuzzy tensor.

Definition 1. (See [25]) An order m dimension n fuzzy tensor $A = (a_{i_{1} \dots i_{m}})$ consists of n^m entries:

$0 \leq a_{i_{1}} \dots_{i_{m}} \leq 1,$ (1) where 1 ≤ i_j ≤ n for j = 1, ⋯ , m.

The set of all nonzero elements of $A$ is denoted by $Φ_{A}$ . Based on the cutting matrix principle of fuzzy matrix [20, 21], we consider the following cutting tensor of fuzzy tensor.

Definition 2. Let $A = (a_{i_{1} \dots i_{m}})$ be a fuzzy tensor of order m and dimension n. For each $λ \in Φ_{A}$ , the definition of the λ level cutting tensor $A_{λ}$ is defined by

$A_{λ} = ((a_{λ})_{i_{1} \dots i_{m}}) = {\begin{matrix} 1, if a_{i_{1} \dots i_{m}} \geq λ, \\ 0, a_{i_{1} \dots i_{m}} < λ . \end{matrix}$ (2)

The definition of cutting tensor is important in studying fuzzy tensor. So hereafter we are not constrained to slice-by-slice notation for expressing the fuzzy tensor. The fuzzy tensor can be expressed easily using more natural cutting tensor notation. Here, we make a definition of scalar product and some basic operation relations of fuzzy tensor, which will be useful in the following.

Definition 3. Let $A = (a_{i_{1} \dots i_{m}})$ be a fuzzy tensor of order m and dimension n. For any λ ∈ (0, 1], the scalar product of λ and $A$ , denoted by $λ A$ , is given by $λ A = (λ \land a_{i_{1} \dots i_{m}})$ .

If $A_{λ}$ is a cutting tensor of fuzzy tensor $A$ , then we have

$λ A_{λ} = (λ \land (a_{λ})_{i_{1} \dots i_{m}}) = {\begin{matrix} λ, if a_{i_{1} \dots i_{m}} \geq λ, \\ 0, a_{i_{1} \dots i_{m}} < λ . \end{matrix}$ (3)

Some basic operation relations between two fuzzy tensors of the same size are as follows.

Definition 4. For arbitrary fuzzy tensor $A = (a_{i_{1} \dots i_{m}})$ and $B = (b_{i_{1} \dots i_{m}})$ of the same size, we say $\begin{matrix} A = B ≜ (a_{i_{1} \dots i_{m}} = b_{i_{1} \dots i_{m}}), \\ A \cup B ≜ (a_{i_{1} \dots i_{m}} \lor b_{i_{1} \dots i_{m}}), \\ A \cap B ≜ (a_{i_{1} \dots i_{m}} \land b_{i_{1} \dots i_{m}}), \\ A^{c} ≜ (1 - a_{i_{1} \dots i_{m}}), \\ A \leq B \Leftrightarrow a_{i_{1} \dots i_{m}} \leq b_{i_{1} \dots i_{m}} . \end{matrix}$

By the above definition we have the following properties of the scalar product.

Property 1. Let $A = (a_{i_{1} \dots i_{m}})$ be a fuzzy tensor of order m and dimension n. If λ₁, λ₂ ∈ (0, 1] and λ₁ ≤ λ₂, then $λ_{1} A \leq λ_{2} A$ .

Proof. Let $λ_{1} A = (c_{i_{1} \dots i_{m}})$ and $λ_{2} A = (d_{i_{1} \dots i_{m}})$ . According to Definition 3, we have c_{i₁⋯i_m} = λ₁ ∧ a_{i₁⋯i_m} and d_{i₁⋯i_m} = λ₂ ∧ a_{i₁⋯i_m}. Note that λ₁ ≤ λ₂, then c_{i₁⋯i_m} = λ₁ ∧ a_{i₁⋯i_m} ≤ λ₂ ∧ a_{i₁⋯i_m} = d_{i₁⋯i_m}. That is, $λ_{1} A \leq λ_{2} A$ .□

Property 2. Let $A = (a_{i_{1} \dots i_{m}})$ and $B = (b_{i_{1} \dots i_{m}})$ are fuzzy tensors of the same size. If $A \leq B$ and λ ∈ (0, 1], then $λ A \leq λ B$ .

Proof. Suppose that there hold $λ A = (c_{i_{1} \dots i_{m}})$ and $λ B = (d_{i_{1} \dots i_{m}})$ . Based on Definition 3, we have c_{i₁⋯i_m} = λ ∧ a_{i₁⋯i_m} and d_{i₁⋯i_m} = λ ∧ b_{i₁⋯i_m}. Because $A \leq B$ , that is, a_{i₁⋯i_m} ≤ b_{i₁⋯i_m}, where c_{i₁⋯i_m} = λ ∧ a_{i₁⋯i_m} ≤ λ ∧ b_{i₁⋯i_m} = d_{i₁⋯i_m}. Hence, $λ A \leq λ B$ .□

Property 3. Let $A = (a_{i_{1} \dots i_{m}})$ and $B = (b_{i_{1} \dots i_{m}})$ are fuzzy tensors of order m and dimension n. If λ ∈ (0, 1], then (1) $λ (A \cup B) = λ A \cup λ B$ , (2) $λ (A \cap B) = λ A \cap λ B$ .

Proof. (1) Let $λ (A \cup B) = (c_{i_{1} \dots i_{m}})$ and $λ A \cup λ B = (d_{i_{1} \dots i_{m}})$ . By Definition 3 and Definition 4, we have $c_{i_{1} \dots i_{m}} = λ \land (a_{i_{1} \dots i_{m}} \lor b_{i_{1} \dots i_{m}})$ and $d_{i_{1} \dots i_{m}} = (λ \land a_{i_{1} \dots i_{m}}) \lor (λ \land b_{i_{1} \dots i_{m}}) .$ Obviously, λ ∧ (a_{i₁⋯i_m} ∨ b_{i₁⋯i_m}) = (λ ∧ a_{i₁⋯i_m}) ∨ (λ ∧ b_{i₁⋯i_m}). That is c_{i₁⋯i_m} = d_{i₁⋯i_m}. Then $λ (A \cup B) = λ A \cup λ B$ . (2) Similar to the proof of (1).□

According to anti-fuzzy theory, a definition and some properties of anti-fuzzy cutting tensor are raised in which the cutting tensor computation is imitated.

Definition 5. Let $A = (a_{i_{1} \dots i_{m}})$ be a fuzzy tensor of order m dimension n. For each $λ \in Φ_{A}$ , the λ level anti-fuzzy cutting tensor $A^{λ}$ is defined by

$A^{λ} = ((a^{λ})_{i_{1} \dots i_{m}}) = {\begin{matrix} 0, if a_{i_{1} \dots i_{m}} \leq λ, \\ 1, a_{i_{1} \dots i_{m}} > λ . \end{matrix}$ (4)

We have the anti-fuzzy scalar product as follows.

Definition 6. Let $A = (a_{i_{1} \dots i_{m}})$ be a fuzzy tensor of order m and dimension n. For any λ ∈ (0, 1], the anti-fuzzy scalar product of λ and $A$ is denoted by $λ \circ A$ , where $λ \circ A = (λ \lor a_{i_{1} \dots i_{m}})$ .

Furthermore, if $A^{λ}$ is an anti-fuzzy cutting tensor of fuzzy tensor $A$ , then

$\begin{matrix} λ \circ A^{λ} = (λ \lor (a^{λ})_{i_{1} \dots i_{m}}) = {\begin{matrix} λ, if a_{i_{1} \dots i_{m}} \leq λ, \\ 1, a_{i_{1} \dots i_{m}} > λ . \end{matrix} \end{matrix}$ (5)

Now, we will only give some properties of the anti-fuzzy scalar product, and the proofs will be omitted.

Property 4. Let $A = (a_{i_{1} \dots i_{m}})$ be a fuzzy tensor of order m and dimension n. If λ₁, λ₂ ∈ (0, 1] and λ₁ ≤ λ₂, then $λ_{1} \circ A \leq λ_{2} \circ A$ .

Property 5. Let $A = (a_{i_{1} \dots i_{m}})$ and $B = (b_{i_{1} \dots i_{m}})$ are fuzzy tensors of the same size. If $A \leq B$ and λ ∈ (0, 1], then $λ \circ A \leq λ \circ B$ .

3 Decomposition theorem of fuzzy tensor

Based on the previous section, we now discuss decomposition theorems of fuzzy tensor, which plays an active role in various fields like the decomposition theorem of tensor and the decomposition theorem of fuzzy matrix. First, we have Decomposition theorem I of fuzzy tensor.

Theorem 1. (Decomposition theorem I) Let $A = (a_{i_{1} \dots i_{m}})$ be a fuzzy tensor of order m and dimension n. For any $λ \in Φ_{A}$ , we have $\begin{matrix} A = ⋃_{λ \in Φ_{A}} λ A_{λ} . \end{matrix}$

Proof. Let $\begin{matrix} ⋃_{λ \in Φ_{A}} λ A_{λ} = B = (b_{i_{1} \dots i_{m}}) . \end{matrix}$ By Definition 2 and Definition 3, we have (3). Note that $\begin{matrix} B = ⋃_{λ \in Φ_{A}} λ A_{λ} = ⋃_{λ \in (0, 1]} λ A_{λ} = ⋁_{λ \in (0, 1]} (λ \land (a_{λ})_{i_{1} \dots i_{m}}) \end{matrix}$ then $\begin{matrix} b_{i_{1} \dots i_{m}} & = ⋁_{λ \in (0, 1]} (λ \land (a_{λ})_{i_{1} \dots i_{m}}) \\ = [⋁_{0 < λ \leq a_{i_{1} \dots i_{m}}} (λ \land (a_{λ})_{i_{1} \dots i_{m}})] \lor \\ [⋁_{a_{i_{1} \dots i_{m}} \leq λ \leq 1} (λ \land (a_{λ})_{i_{1} \dots i_{m}})] \\ = (⋁_{0 < λ \leq a_{i_{1} \dots i_{m}}} λ) \lor (⋁_{a_{i_{1} \dots i_{m}} \leq λ \leq 1} 0) \\ = a_{i_{1} \dots i_{m}} . \end{matrix}$ Therefore, $\begin{matrix} A = ⋃_{λ \in Φ_{A}} λ A_{λ} . \end{matrix}$

□

Now, from Decomposition theorem I, we know that the fuzzy tensor is uniquely expressed by cutting tensor. This fact results in the following algorithm.

An algorithm to decomposition of fuzzy tensor. Step 1. Given a fuzzy tensor $A = (a_{i_{1} \dots i_{m}})$ . Step 2. Find out all of nonzero elements of $A$ , which are written in sets $Φ_{A}$ . Step 3. Compute all of cutting tensor, for each $λ \in Φ_{A}$ . Step 4. According to Decomposition theorem I, output the decomposition forms of fuzzy tensor $A$ . To demonstrate this algorithm, we make the following numerical example for decomposing fuzzy tensor.

Example 1. Consider $A = (a_{ijk})$ , a fuzzy tensor of order 3 and dimension 2, where $A = [A (1, :, :), A (2, :, :)]$ and $A (1, :, :) = (\begin{matrix} 0.6 & 0.5 \\ 0.4 & 0.3 \end{matrix})$ , $A (2, :, :) = (\begin{matrix} 0.7 & 0.6 \\ 0.5 & 0.4 \end{matrix})$ . Find out $Φ_{A} = {0.7, 0.6, 0.5, 0.4, 0.3}$ . By calculating $A_{λ}$ for $λ \in Φ_{A}$ , we get Table 1.

Table 1
Numerical data for example $A_{λ}$

λ A_λ (1,: ,:) A_λ (2,: ,:)

0.7 $(\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix})$ $(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})$

0.6 $(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})$ $(\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix})$

0.5 $(\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix})$ $(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix})$

0.4 $(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix})$ $(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})$

0.3 $(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})$ $(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})$

So $A (1, :, :) = 0.7 A_{0.7} (1, :, :) \cup 0.6 A_{0.6} (1, :, :) \cup 0.5 A_{0.5} (1, :, :) \cup 0.4 A_{0.4} (1, :, :) \cup 0.3 A_{0.3} (1, :, :) = (\begin{matrix} 0.6 \lor 0.5 \lor 0.4 \lor 0.3 & 0.5 \lor 0.4 \lor 0.3 \\ 0.4 \lor 0.3 & 0.3 \end{matrix}) = (\begin{matrix} 0.6 & 0.5 \\ 0.4 & 0.3 \end{matrix})$ , A (2,: ,:) =0.7A_0.7 (2,: ,:) ∪0.6A_0.6 (2,: ,:) ∪0.5A_0.5 (2,: ,:) ∪0.4A_0.4 (2,: ,:) ∪0.3A_0.3 (2,: ,:) $= (\begin{matrix} 0.7 \lor 0.6 \lor 0.5 \lor 0.4 \lor 0.3 & 0.6 \lor 0.5 \lor 0.4 \lor 0.3 \\ 0.5 \lor 0.4 \lor 0.3 & 0.4 \lor 0.3 \end{matrix}) = (\begin{matrix} 0.7 & 0.6 \\ 0.5 & 0.4 \end{matrix})$ .

That is, $\begin{matrix} A = 0.7 A_{0.7} \cup 0.6 A_{0.6} \cup 0.5 A_{0.5} \cup 0.4 A_{0.4} \cup 0.3 A_{0.3} \\ = ⋃_{λ \in Φ_{A}} λ A_{λ} . \end{matrix}$

Example 1, we see explicitly that the fuzzy tensor can be shown by its own elements and the corresponding cutting tensor. So, we affirm that decomposition of fuzzy tensor depends heavily on the elements contained in itself. Furthermore, from the proof of Decomposition theorem I, we obtain easily the following results.

Corollary 1. Let $A = (a_{i_{1} \dots i_{m}})$ be a fuzzy tensor of order m and dimension n. We have $\begin{matrix} A = ⋁_{λ \in Φ_{A}} (λ \land (a_{λ})_{i_{1} \dots i_{m}}) . \end{matrix}$

Corollary 2. Let $A = (a_{i_{1} \dots i_{m}})$ be a fuzzy tensor of order m and dimension n. We will give every element of fuzzy tensor $A$ as follows: a_{i₁⋯i_m} = ⋁ {λ ∈ [0, 1] : (a_λ) _{i₁⋯i_m} ≠ 0}.

From Corollary 2, if all of cutting tensors of a fuzzy tensor are known, then the fuzzy tensor is easily obtained. To study further in this area, we shall describe decomposition of fuzzy tensor through an anti-fuzzy ideas. Using the idea of anti-fuzzy, we also acquire the corresponding decomposition theorem of fuzzy tensor whose proof is similar in fuzzy conditions. Therefore, we only expand the conclusions of decomposition theorem under anti-fuzzy conditions.

Theorem 2. (Decomposition theorem II) Let $A = (a_{i_{1} \dots i_{m}})$ be a fuzzy tensor of order m and dimension n. For any $λ \in Φ_{A}$ , we have $\begin{matrix} A = ⋂_{λ \in Φ_{A}} (λ \circ A^{λ}) . \end{matrix}$

Proof. Let $\begin{matrix} ⋂_{λ \in Φ_{A}} (λ \circ A^{λ}) = B = (b_{i_{1} \dots i_{m}}) . \end{matrix}$ By Definition 5 and Definition 6, we have (5). Note that $\begin{matrix} B = ⋂_{λ \in Φ_{A}} λ \circ A^{λ} = ⋂_{λ \in (0, 1]} λ \circ A^{λ} \\ = ⋀_{λ \in (0, 1]} (λ \lor (a^{λ})_{i_{1} \dots i_{m}}) \end{matrix}$ then $\begin{matrix} b_{i_{1} \dots i_{m}} & = ⋀_{λ \in (0, 1]} (λ \lor (a^{λ})_{i_{1} \dots i_{m}}) \\ = [⋀_{0 < λ \leq a_{i_{1} \dots i_{m}}} (λ \lor (a^{λ})_{i_{1} \dots i_{m}})] \\ \land [⋀_{a_{i_{1} \dots i_{m}} \leq λ \leq 1} (λ \lor (a^{λ})_{i_{1} \dots i_{m}})] \\ = (⋀_{0 < λ \leq a_{i_{1} \dots i_{m}}} 1) \land (⋀_{a_{i_{1} \dots i_{m}} \leq λ \leq 1} λ) \\ = a_{i_{1} \dots i_{m}} . \end{matrix}$ So $\begin{matrix} A = ⋂_{λ \in Φ_{A}} (λ \circ A^{λ}) . \end{matrix}$

□

Using Theorem 2 for Example 1. Every $λ \in Φ_{A}$ , computing $A^{λ}$ . Here A (1,: ,:) =0.7 ∘ A^0.7 (1,: ,:) ∩0.6 ∘ A^0.6 (1,: ,:) ∩ 0.5 ∘ A^0.5 (1,: ,:) ∩0.4 ∘ A^0.4 (1,: ,:) ∩ 0.3 ∘ A^0.3 (1,: ,:) $= (\begin{matrix} 0.7 \land 0.6 \land 1 \land 1 \land 1 & 0.7 \land 0.6 \land 0.5 \land 1 \land 1 \\ 0.7 \land 0.6 \land 0.5 \land 0.4 \land 1 & 0.7 \land 0.6 \land 0.5 \land 0.4 \land 0.3 \end{matrix}) = (\begin{matrix} 0.6 & 0.5 \\ 0.4 & 0.3 \end{matrix})$ , A (2,: ,:) =0.7 ∘ A^0.7 (2,: ,:) ∩ 0.6 ∘ A^0.6 (2,: ,:) ∩0.5 ∘ A^0.5 (2,: ,:) ∩ 0.4 ∘ A^0.4 (2,: ,:) ∩0.3 ∘ A^0.3 (2,: ,:) $= (\begin{matrix} 0.7 \land 1 \land 1 \land 1 \land 1 & 0.7 \land 0.6 \land 1 \land 1 \land 1 \\ 0.7 \land 0.6 \land 0.5 \land 1 \land 1 & 0.7 \land 0.6 \land 0.5 \land 0.4 \land 1 \end{matrix}) = (\begin{matrix} 0.7 & 0.6 \\ 0.5 & 0.4 \end{matrix})$ . That is, $\begin{matrix} A = 0.7 \circ A^{0.7} \cap 0.6 \circ A^{0.6} \cap 0.5 \circ A^{0.5} \\ \cap 0.4 \circ A^{0.4} \cap 0.3 \circ A^{0.3} \\ = ⋂_{λ \in Φ_{A}} λ \circ A^{λ} . \end{matrix}$

From above, it can be illustrated that a fuzzy tensor can be decomposed by the same degree as Theorem 1 and Theorem 2. In practical problem, according to the actual situation, we choose an application from decomposition theorem.

Fuzzy tensor decomposition theorem I shows that a fuzzy set $A$ is determined by $A_{λ}$ using scalar product of fuzzy state and fuzzy tensor decomposition theorem II shows that a fuzzy set $A$ is determined by $A^{λ}$ using anti-fuzzy scalar product of anti-fuzzy condition. The two decomposition theorems of the fuzzy tensor are mutually complementary. In practical problems, the decomposition theorem is chosen from two fuzzy tensor decomposition theorems according to actual demands to solve practical problems. After the decomposition theorem I and the decomposition theorem II, for multi-objective decision, multi-dimentional optimization of fuzzy decision, multi-dimentional fuzzy system engineering, cluster fuzzy analysis, scheduling theory, statistical theory, and so on, according to actual demands we can choose proper decomposition theorem for dealing with practical problems.

4 Real applications of decomposition theorem

We consider an application related to National Standard of Congou black tea, that is, National Standard of the People’s Republic of China GB/T 13738.2–2008. According to the varieties of tea and the requirements of the products, two products having large leaves and small leaves are obtained. The requirements of national standards include basic requirements, sensory quality, physical and chemical indexes, hygienic indexes and net contents.

The water extracts is a very important physical and chemical index of Congou black tea. The physical and chemical indexes of the national standard water extracts are shown in the following Table 2. Our samples are dried specimens made from fresh leaves of old tree tea from the People’s Republic of China, Pu’an County of Guizhou province: Pu’an No.1, Pu’an No.2 and Pu’an No.3. They belong to the large leaf species. Three samples are obtained at 80°C, 90°C and 100°C, and the extraction time was 25 min, 35 min and 45 min, respectively. The water extracts data of three kinds of samples at three temperatures and three extraction time were listed in Table 3.

Table 2
Water extracts index

item Special - 1 grade 2 - 3 grade 4 - 6 grade

Large leaves ≥36 ≥34 ≥32

Small leaves ≥32 ≥30 ≥28

Table 3

Water extracts data

	25min	35min	45min
80°C	$(\begin{matrix} 19.70 & 20.21 & 22.53 \end{matrix})$	$(\begin{matrix} 22.31 & 21.32 & 24.63 \end{matrix})$	$(\begin{matrix} 26.85 & 25.65 & 25.42 \end{matrix})$
90°C	$(\begin{matrix} 20.83 & 24.83 & 26.89 \end{matrix})$	$(\begin{matrix} 24.52 & 25.42 & 28.12 \end{matrix})$	$(\begin{matrix} 29.71 & 27.86 & 28.65 \end{matrix})$
100°C	$(\begin{matrix} 25.48 & 26.70 & 28.75 \end{matrix})$	$(\begin{matrix} 32.80 & 32.91 & 33.57 \end{matrix})$	$(\begin{matrix} 34.78 & 34.43 & 34.45 \end{matrix})$

For each data divided by 100, the fuzzy data are represented as $A (1, :, :)$ , $A (2, :, :)$ and $A (3, :, :)$ , respectively, in Table 4. A fuzzy threshold is obtained by dividing the water extracts index value by 100. By the fuzzy tensors decomposition theorem I, for fuzzy threshold α = 0.34 and α = 0.32, we can get three kinds of tribute tea samples which have reached four–six grade at 35 min and in 100°C and two–three grade at 45 min and in 100°C. According to this application, we can consider that the extraction temperature has obvious influence on tea water extracts, so the extraction temperature should be increased as much as possible to improve the extraction content. The extraction time also affected the tea water extracts, but it had little effect on the extraction temperature. Therefore, the extraction temperature should be 100°C, and the extraction time should be 35 min – 45 min. Through the example, we can summarize the advantages of fuzzy cutting set theory using fuzzy tensor decomposition, one of which is to solve some problems in tensor decomposition, such as difficulty in rank calculation and too large-scale computation. The second is to solve the fuzzy matrix decomposition only for one sample.

Table 4

$A (1, :, :)$ , $A (2, :, :)$ and $A (3, :, :)$ data

$A (1, :, :)$	$A (2, :, :)$	$A (3, :, :)$
$(\begin{matrix} 0.1970 & 0.2231 & 0.2685 \\ 0.2083 & 0.2452 & 0.2971 \\ 0.2548 & 0.3280 & 0.3478 \end{matrix})$	$(\begin{matrix} 0.2021 & 0.2132 & 0.2565 \\ 0.2483 & 0.2542 & 0.2786 \\ 0.2670 & 0.3291 & 0.3443 \end{matrix})$	$(\begin{matrix} 0.2253 & 0.2463 & 0.2542 \\ 0.2689 & 0.2812 & 0.2865 \\ 0.2875 & 0.3357 & 0.3445 \end{matrix})$

5 Conclusions

In this paper, we deliberate decomposition theory of fuzzy tensor by cutting tensors and scalar product. We emphatically canvass and prove decomposition theorem I of fuzzy tensor. Our numerical example and applications show that the algorithm of decomposing fuzzy tensor is feasible and effective. Furthermore, we discuss decomposition theorem II of fuzzy tensor by anti-cutting tensor and anti-fuzzy scalar product. However, there are still many problems in practice needed to concern about. In the future, we will continue to research application problems of fuzzy tensor.

Footnotes

Acknowledgments

This work was supported by the Doctor Foundation of Shandong Jianzhu University and National Natural Science Foundation of China (Grant No.11671105 and No.11461018).

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λ	A_λ (1,: ,:)	A_λ (2,: ,:)
0.7	$(\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix})$	$(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})$
0.6	$(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})$	$(\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix})$
0.5	$(\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix})$	$(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix})$
0.4	$(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix})$	$(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})$
0.3	$(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})$	$(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})$

item	Special - 1 grade	2 - 3 grade	4 - 6 grade
Large leaves	≥36	≥34	≥32
Small leaves	≥32	≥30	≥28

Decomposition theorem of fuzzy tensors and its applications

Abstract

Keywords

1 Introduction

2 Fuzzy tensor and cutting tensor

Table 1 Numerical data for example A λ λ A λ (1,: ,:) A λ (2,: ,:) 0.7 ( 0 0 0 0 ) ( 1 0 0 0 ) 0.6 ( 1 0 0 0 ) ( 1 1 0 0 ) 0.5 ( 1 1 0 0 ) ( 1 1 1 0 ) 0.4 ( 1 1 1 0 ) ( 1 1 1 1 ) 0.3 ( 1 1 1 1 ) ( 1 1 1 1 )

Table 2 Water extracts index item Special - 1 grade 2 - 3 grade 4 - 6 grade Large leaves ≥36 ≥34 ≥32 Small leaves ≥32 ≥30 ≥28

Footnotes

Acknowledgments

References

Table 2
Water extracts index

item Special - 1 grade 2 - 3 grade 4 - 6 grade

Large leaves ≥36 ≥34 ≥32

Small leaves ≥32 ≥30 ≥28