Approximation of fuzzy neural networks based on Choquet integral

Abstract

In this paper, we shall introduce a metric on the space of fuzzy-valued measurable functions by means of a kind of nonlinear integral — Choquet integral with respect to a capacity. We called it as Choquet integral norm. By using the metric between fuzzy-valued measurable functions, we discuss the mean approximation of regular fuzzy neural networks in the sense of Choquet integral. It is shown that any fuzzy-valued measurable function on $ℝ$ can be approximated in mean by four-layer regular fuzzy neural networks in the sense of Choquet integral norm.

Keywords

Capacity Choquet integral regular fuzzy neural network submodularity

1 Introduction

The learning ability of a neural network is closely related to its approximating capabilities. So it is important and interesting to study the approximation properties of neural networks. The studies on this matter were undertaken by many authors and a great number of important results were obtained ([1 , 37] etc.). The similar approximation problems in fuzzy environment were investigated by Buckley [2, 3], Liu [32 –34], and Li et al. [27, 28]. Castillo and Castro et al. investigated interval type-2 fuzzy neural networks and type-2 fuzzy neural networks, and obtained a lot of meaningful results [5 –10]. In [33], it is proved that a continuous fuzzy-valued function can be closely approximated by a class of regular fuzzy neural networks (RFNNs) with real input and fuzzy-valued output. In [27, 28] Li et al. discussed mean approximation of four-layer RFNNs in the sense of Sugeno integral (it is also known as fuzzy integral) on finite continuous fuzzy measure spaces.

In this paper, we shall discuss the mean approximation of fuzzy neural network in the sense of Choquet integral. In our discussion the capacities related to Choquet integral are supposed to be submodular [35] and fulfil the condition (E) [29]. Under such conditions we shall prove that RFNNs is a pan-approximator based on Choquet integral for fuzzy-valued measurable function. That is, any fuzzy-valued measurable function can be meanly approximated by the four-layer RFNNs in the sense of Choquet integral. The previous researches on this topics made by Li et al. [27, 28] are extended.

2 Preliminaries

Let $ℝ$ be the set of all real numbers, and $B$ denote Borel σ-algebra on $ℝ$ . $(ℝ, B)$ denotes Borel measurable space. Unless stated otherwise all the subsets mentioned are supposed to belong to $B$ .

Definition 2.1. [38] A set function $μ : B \to [0, \infty]$ is called a monotone measure, if it satisfies the following properties:

μ (∅) =0;

A ⊂ B implies μ (A) ≤ μ (B) (monotonicity).

A monotone measure μ is called a capacity if it satisfies $μ (ℝ) = 1$ .

In some literature, a monotone measure is also known a fuzzy measure or a non-aditive measure or non-additive probability (see [15 , 38])

In this paper, we always assume that μ is a capacity in the sense of Definition 2.1.

Consider a nonnegative real-valued measurable function f on $A \in B$ , the Choquet integral of f on A with respect to a capacity μ, denoted by (C) ∫_Afdμ, is defined by $(C) \int_{A} f d μ = \int_{0}^{\infty} μ ({x : f (x) \geq t} \cap A) dt,$ where the right side integral is Riemann integral [14] (see aso [35, 38]).

Definition 2.2. [35] A capacity μ is said to be submodular, if for any $A, B \in B$ , we have $μ (A \cup B) + μ (A \cap B) \leq μ (A) + μ (B) .$ Note that Choquet integral is a type of non-linear integral, in general, $(C) \int_{A} (f + g) d μ \neq (C) \int_{A} f d μ + (C) \int_{A} g d μ .$ We have the following result.

Proposition 2.1. [35] Let μ be a capacity and f, g be nonnegative real-valued measurable functions on $A \in B$ . If μ is submodular, then $(C) \int_{A} (f + g) d μ \leq (C) \int_{A} f d μ + (C) \int_{A} g d μ .$

Definition 2.3. A capacity $μ : B \to [0, 1]$ is said to be

continuous from below, if $lim_{n \to \infty} μ (A_{n}) = μ (A)$ whenever A_n ↗ A;

continuous from above, if $lim_{n \to \infty} μ (A_{n}) = μ (A)$ whenever A_n ↘ A;

continuous, if μ is both continuous from below and from above;

strongly order continuous [29], if $lim_{n \to \infty}$ μ (A_n)=0 whenever A_n↘A with μ (A)=0;

null-additive [38], if μ (E ∪ F) = μ (E) for any E whenever μ (F) =0;

weakly null-additive [38], if for any $E, F \in B$ , μ (E) = μ (F) =0 imply μ (E ∪ F) =0;

weakly asymptotic null-additive [19], if for any decreasing sequences {E_n} and {F_n} with μ (E_n) ↘0 (n → ∞) and μ (F_n) ↘0 (n → ∞), we have μ (E_n ∪ F_n) →0 (n → ∞).

Obviously, the null-additivity of μ implies weak null-additivity.

Definition 2.4. A capacity μ is said to have

pseudometric generating property (for short, (p.g.p.)) [17], if for any ${E_{n}} \subset B$ and ${F_{n}} \subset B$ ,

μ (E_{n}) \lor μ (F_{n}) \to 0 \Rightarrow μ (E_{n} \cup F_{n}) \to 0;

property (S) [29], if for any ${A_{n}}_{n} \subset B$ with $lim_{n \to + \infty} μ (A_{n}) = 0$ , there exists a subsequence {A_{n
_i}} _i of {A_n} _n such that $μ (⋂_{k = 1}^{\infty} ⋃_{i = k}^{\infty} A_{n_{i}}) = 0$ .

From Definition 2.2, 2.3 and 2.4, we have the following propoerty.

Proposition 2.2. If μ is submodular, then it has (p.g.p.) and while the later implies weak null-additivity and weak asymptotic null-additivity.

Definition 2.5. [29] A capacity μ is said to fulfil the condition (E), if for any ɛ > 0 and any double sequence ${A_{n}^{(k)} ∣ n \geq 1, k \geq 1} \subset B$ satisfying $A_{n}^{(k)} ↘ D_{n} (k \to \infty), n = 1, 2, \dots$ and $μ (⋃_{n = 1}^{\infty} D_{n}) = 0$ , there exists a subsequence ${A_{n}^{(k_{n})}}$ of ${A_{n}^{(k)} ∣ n \geq 1, k \geq 1}$ such that $μ (⋃_{n = 1}^{\infty} A_{n}^{(k_{n})}) < ɛ (k_{1} < k_{2} < \dots) .$

Proposition 2.3. [29] If μ is continuous, i.e. it is both continuous from below and from above, then μ fulfils condition (E).

Proposition 2.4. [21–23 , 25] If μ is strongly order continuous and has property (S), then μ fulfils condition (E).

3 The regularity of capacity

The regularity of a capacity plays an important role in our discussion.

Definition 3.1. [35, 39] A capacity μ defined on $B$ is called regular, if for each $A \in B$ and each ɛ > 0, there exist a closed set $F_{ɛ} \in B$ and an open set $G_{ɛ} \in B$ such that $F_{ɛ} \subset A \subset G_{ɛ} and μ (G_{ɛ} - F_{ɛ}) < ɛ .$

Proposition 3.1. If μ is regular, then for each $A \in B$ and each ɛ > 0, there exist a closed set $F_{ɛ} \in B$ such thatF_ɛ ⊂ A and μ (A - F_ɛ) < ɛ.

The following results present some sufficient conditions for regularity (see [18 , 30])

Proposition 3.2. If μ satisfies one of the following conditions, then μ is regular.

μ is continuous and weakly null-additive;

μ is continuous and null-additive;

μ is continuous and, it has (p.g.p.);

μ is continuous and weakly asymptotic null-additive;

μ fulfils the condition (E) and has (p.g.p.);

μ is strongly order continuous, weakly null-additive and has property (S).

A real-valued function φ (x) defined on $X \in B$ is called a simple function, if $φ (x) = \sum_{n = 1}^{s} c_{k} χ_{E_{n}} (x) (x \in X),$ where χ_{E
_n} (x) is the characteristic function of the set $E_{n} \in B$ , i.e., $χ_{E_{n}} (x) = {\begin{matrix} 1, & if x \in E_{n}; \\ 0, & if x \in X - E_{n}, \end{matrix}$ and $X = ⋃_{n = 1}^{s} E_{n}$ (a disjoint finite union).

In the following, we give a characteristic of simple function for a capacity. We call it as Lusin type of theorem for a capacity.

Theorem 3.1. (Lusin-like theorem) Let μ be regular and weakly asymptotic null-additive. If φ is a simple function on X, then, for every ɛ > 0, there exists a closed subset F_ɛ ⊂ X such that φ is continuous on F_ɛ and μ (X - F_ɛ) < ɛ .

Proof. Let

$φ (x) = \sum_{n = 1}^{s} c_{k} χ_{E_{n}} (x) (x \in X),$ where χ_{E
_n} (x) is the characteristic function of the set E_n and $X = ⋃_{n = 1}^{s} E_{n}$ (a disjoint finite union). For every fixed E_n (n = 1, 2, …, s), by Proposition 3.1, there exists a sequence ${F_{n}^{(k)}}_{k = 1}^{\infty}$ of closed sets such that $F_{n}^{(k)} \subset E_{n} and μ (E_{n} - F_{n}^{(k)}) < \frac{1}{k}$ for any k = 1, 2, …. We may assume that ${F_{n}^{(k)}}_{k = 1}^{\infty}$ is increasing in k for each fixed n, without any loss of generality. Thus for each fixed n, ${E_{n} - F_{n}^{(k)}}$ is decreasing in k and $lim_{k \to \infty} μ (E_{n} - F_{n}^{(k)}) = 0 .$ By using the weak asymptotic null-additivity of μ, we have $lim_{k \to \infty} μ (⋃_{n = 1}^{s} (E_{n} - F_{n}^{(k)})) = 0 .$ Therefore, for any ɛ > 0, there exists k₀ such that $μ (⋃_{n = 1}^{s} (E_{n} - F_{n}^{(k_{0})})) < ɛ .$

Put $F_{ɛ} = ⋃_{n = 1}^{s} F_{n}^{(k_{0})}$ , then φ is continuous on the closed subset $F_{ɛ} \in B$ , and noting that $⋃_{n = 1}^{s} E_{n} - ⋃_{n = 1}^{s} F_{n}^{(k_{0})} \subset ⋃_{n = 1}^{s} (E_{n} - F_{n}^{(k_{0})}),$ we have $\begin{matrix} μ (X - F_{ɛ}) & \leq & μ (⋃_{n = 1}^{s} E_{n} - ⋃_{n = 1}^{s} F_{n}^{(k_{0})}) \\ \leq & μ (⋃_{n = 1}^{s} (E_{n} - F_{n}^{(k_{0})})) \\ < & ɛ . □ \end{matrix}$

From Proposition 2.2, 3.2 and Theorem 3.1, we can obtain the following results.

Corollary 3.1. If μ satisfies the one of conditions (1)–(6) in Proposition 3.2, then the conclusion of Theorem 3.1 holds.

Corollary 3.2. Let μ be submodular. If μ fulfils the Condition (E), then the conclusion of Theorem 3.1 holds.

Corollary 3.3. Let μ be submodular. If μ is continuous, then the conclusion of Theorem 3.1 holds.

Corollary 3.4. Let μ be strongly order continuous and have property (S). If μ is weakly null-additive, then Theorem 3.1 holds (see [20 , 29]).

4 Approximation in fuzzy mean by regular fuzzy neural networks

In this section, we study an approximation property of the four-layer RFNNs to fuzzy-valued measurable function in the sense of mean convergence based on Choquet integral.

Let $F_{0} (ℝ)$ be the set of all bounded fuzzy numbers, i.e., for $\tilde{A} \in F_{0} (ℝ)$ , the following conditions hold:

$\forall λ \in (0, 1], {\tilde{A}}_{λ} = {x \in ℝ ∣ \tilde{A} (x) \geq λ}$ is the closed interval of $ℝ$ ;

Supp $(\tilde{A}) = \bar{{x \in ℝ ∣ \tilde{A} (x) > 0}} \subset [0, 1]$ ;

${x \in ℝ ∣ \tilde{A} (x) = 1} \neq \emptyset$ .

The above-mentioned bounded fuzzy number is called fuzzy value (see [11 –13]).

For $\tilde{A}, \tilde{B} \in F_{0} (ℝ)$ , define metric $D (\tilde{A}, \tilde{B})$ between $\tilde{A}$ and $\tilde{B}$ by $D (\tilde{A}, \tilde{B}) = sup_{λ \in [0, 1]} d_{H} ({\tilde{A}}_{λ}, {\tilde{B}}_{λ})$ where d_H means Hausdorff metric: for $A, B \subset ℝ$ , $\begin{matrix} d_{H} (A, B) \\ = max {sup_{x \in A} inf_{y \in B} (| x - y |), sup_{y \in B} inf_{x \in A} (| x - y |)} . \end{matrix}$

It is known that $(F_{0} (ℝ), D)$ is a completely separable metric space [16].

For simplicity, supp( $\tilde{A}$ ) is also written as ${\tilde{A}}_{0}$ . Obviously, ${\tilde{A}}_{0}$ is a bounded and closed interval in [1]. For $\tilde{A} \in F_{0} (ℝ)$ , let ${\tilde{A}}_{λ} = [a_{λ}^{-}, a_{λ}^{+}]$ for each λ ∈ [0, 1], then $[a_{λ}^{-}, a_{λ}^{+}] \subset [0, 1]$ . We denote $| \tilde{A} | = sup_{λ \in [0, 1]} a_{λ}^{+}$ , then $| \tilde{A} | \leq 1$ .

Proposition 4.1. [33] Assume $\tilde{A}, \tilde{A_{1}}, \tilde{A_{2}} \in F_{0} (ℝ)$ , and $\tilde{W_{i}}, \tilde{V_{i}} \in F_{0} (ℝ) (i = 1, 2, \dots, n)$ . Then

$D (\tilde{A} \cdot \tilde{A_{1}}, \tilde{A} \cdot \tilde{A_{2}}) \leq | \tilde{A} | \cdot D (\tilde{A_{1}}, \tilde{A_{2}})$ ,

$D (\sum_{i = 1}^{n} {\tilde{W}}_{i}, \sum_{i = 1}^{n} {\tilde{V}}_{i}) \leq \sum_{i = 1}^{n} D ({\tilde{W}}_{i}, {\tilde{V}}_{i})$ .

Let X be a measurable set in $ℝ$ . $L (X)$ denotes the set of all fuzzy-valued measurable functions $\tilde{F} : X \to F_{0} (ℝ) .$

For any $\tilde{F}, \tilde{G} \in L (X)$ , $D (\tilde{F}, \tilde{G})$ is measurable function on $(X, B \cap X)$ .

We define $D_{Ch} (\tilde{F}, \tilde{G}) = (C) \int_{X} D (\tilde{F}, \tilde{G}) d μ .$

Then 𝔇_Ch is called Choquet integral norm.

Since D is metric defined on $F_{0} (ℝ)$ , by Proposotion 2.1, we have the following result.

Proposition 4.2. If the capacity μ is submodular, then for any $\tilde{F}, \tilde{G}, \tilde{H} \in L (X)$ , $𝔇_{Ch} (\tilde{F}, \tilde{H}) \leq 𝔇_{Ch} (\tilde{F}, \tilde{G}) + 𝔇_{Ch} (\tilde{G}, \tilde{H}) .$ Noting that for any $x \in X, D (\tilde{F}, \tilde{G}) \leq 1$ , we have $\begin{matrix} \tilde{G}) & = & (C) \int_{X} D (\tilde{F}, \tilde{G}) d μ \\ = & \int_{0}^{\infty} μ ({x : D (\tilde{F}, \tilde{G}) \geq t} \cap X) dt \\ = & \int_{0}^{1} μ ({x : D (\tilde{F}, \tilde{G}) \geq t} \cap X) dt . \end{matrix}$

Definition 4.1 Let $\tilde{F}, {\tilde{F}}_{n} (n = 1, 2, \dots) \in L (X)$ . We say that a sequence of fuzzy-valued measurable functions ${{\tilde{F}}_{n}}$ convergences in mean to $\tilde{F}$ in the sense of the Choquet integral, denoted by ${\tilde{F}}_{n} \overset{c . m .}{⟶} \tilde{F}$ , if $lim_{n \to + \infty} (C) \int_{X} D ({\tilde{F}}_{n}, \tilde{F}) d μ = 0 .$

Definition 4.2. [33] A fuzzy-valued function $\tilde{Φ} : X \to F_{0} (ℝ)$ is called a fuzzy-valued simple function, if there exist ${\tilde{A}}_{1}, {\tilde{A}}_{2}, \dots, {\tilde{A}}_{m} \in F_{0} (ℝ)$ , such that ∀ x ∈ X, $\tilde{Φ} (x) = \sum_{k = 1}^{m} χ_{X_{k}} (x) \cdot {\tilde{A}}_{k},$ where $X_{k} \in B \cap X (k = 1, 2, \dots, m), X_{i} \cap X_{j} = \emptyset (i \neq j)$ and $X = ⋃_{k = 1}^{m} X_{k}$ , χ_{X
_k} (x) is the characteristic function of the set X_k.

Denotes $S (X)$ the set of all fuzzy-valued simple functions, then $S (X) \subset L (X)$ .

Proposition 4.3. If $\tilde{F} \in L (X)$ , then for every ɛ > 0, there exists ${\tilde{Φ}}_{ɛ} \in S (X)$ such that $𝔇_{Ch} (\tilde{F}, {\tilde{Φ}}_{ɛ}) < ɛ .$

In the following we recall the definition of four-layer feedforward RFNN (cf. [33]).

Define $\begin{matrix} H [σ] = \\ {\tilde{H} | \tilde{H} (x) = \sum_{i = 1}^{n} {\tilde{W}}_{i} \cdot (\sum_{j = 1}^{m} v_{ij} \cdot σ (x \cdot {\tilde{u}}_{j} + {\tilde{Θ}}_{j}))} . \end{matrix}$ where σ is a given extended function of $σ : ℝ \to ℝ$ (bounded, continuous and nonconstant), and $x \in ℝ, {\tilde{W}}_{i}, {\tilde{V}}_{ij}, {\tilde{U}}_{j}, {\tilde{Θ}}_{j} \in F_{0} (ℝ)$ .

For any $\tilde{H} \in H [σ]$ , $\tilde{H}$ is a four-layer feedforward RFNN with activation function σ, threshold vector $({\tilde{Θ}}_{1}, \dots, {\tilde{Θ}}_{m})$ in the first hidden layer (see Fig. 4.1).

Restricting fuzzy numbers ${\tilde{V}}_{ij}, {\tilde{U}}_{j}, {\tilde{Θ}}_{j} \in F_{0} (ℝ)$ , respectively, to be real numbers $v_{ij}, u_{j}, Θ_{j} \in ℝ$ , we obtain the subset $H_{0} [σ]$ of $H [σ]$ : $\begin{matrix} H_{0} [σ] = \\ {\tilde{H} | \tilde{H} (x) = \sum_{i = 1}^{n} {\tilde{W}}_{i} \cdot (\sum_{j = 1}^{m} v_{ij} \cdot σ (x \cdot u_{j} + θ_{j}))} . \end{matrix}$

Definition 4.3.

$H_{0} [σ]$ is called the pan-approximator of $S (X)$ in the sense of DD_Ch, if $\forall \tilde{Φ} \in S (X)$ , ∀ ɛ > 0, there exists ${\tilde{H}}_{ɛ} \in H_{0} [σ]$ such that

𝔇_{Ch} (\tilde{Φ}, {\tilde{H}}_{ɛ}) < ɛ .

For $\tilde{F} \in L (X)$ , $H [σ]$ is called the pan-approximator for $\tilde{F}$ in the sense of DD_Ch, if ∀ ɛ > 0, there exists ${\tilde{H}}_{ɛ} \in H [σ]$ such that

𝔇_{Ch} (\tilde{F}, {\tilde{H}}_{ɛ}) < ɛ .

By using Lusin-like theorem (Theorem 3.1), we shall prove the following theorem, which is the main result in this paper.

Theorem 4.1. Let μ be a submodular capacity and fulfil the condition (E). Then,

$H_{0} [σ]$ is the pan-approximator of $S (X)$ in the sense of DD_Ch.

$H [σ]$ is the pan-approximator for $\tilde{F}$ in the sense of DD_Ch.

Proof. By using the conclusion of (1) and Proposition 4.3 we can obtain (2). Now we only prove (1).

Suppose that $\tilde{Φ} (x)$ is a fuzzy-valued simple function, i.e., $\tilde{Φ} (x) = \sum_{k = 1}^{m} χ_{X_{k}} (x) \cdot {\tilde{A}}_{k} (x \in X) .$

For arbitrarily given ɛ > 0, applying Theorem 3.1 (Lusin-like theorem) to each real measurable function χ_{X
_k} (x), for every fixed k (1 ≤ k ≤ m), there exists closed set $F_{k} \in B \cap X$ such that $F_{k} \subset X_{k} and μ (X_{k} - F_{k}) < \frac{ɛ}{2}$ and χ_{X
_k} (x) is continuous on F_k. Therefore, for every k there exist a Tauber-Wiener function σ and $p_{k} \in N, v_{k1}^{'}, v_{k2}^{'}, \dots, v_{kpk}^{'}$ , $θ_{k1}^{'}, θ_{k2}^{'}, \dots, θ_{kpk}^{'} \in ℝ$ , and $w_{k1}^{'}$ , $w_{k2}^{'}$ , ⋯, $w_{kpk}^{'}$ ∈ $ℝ^{n}$ such that for any x ∈ F_k, $\begin{matrix} | χ_{X_{k}} (x) - \sum_{j = 1}^{p_{k}} v_{kj}^{'} \cdot σ (〈 w_{kj}^{'}, x 〉 + θ_{kj}^{'}) | \\ < \frac{ɛ}{2 \sum_{k = 1}^{m} | {\tilde{A}}_{k} |} . \end{matrix}$ (We can assume $\sum_{k = 1}^{m} | {\tilde{A}}_{k} | \neq 0$ , without any loss of generality).

Denote $F = ⋃_{k = 1}^{m} F_{k}$ , then X = F ∪ (X - F). Since μ is submodular, it has (p.g.p.) (Proposition 2.2). By the condition (E) and the (p.g.p.) of capacity μ, we can take the family of closed sets {F₁, F₂, ⋯ , F_m} such that $\begin{matrix} μ (X - F) & = & μ (⋃_{k = 1}^{m} X_{k} - ⋃_{k = 1}^{m} F_{k}) \\ \leq & μ (⋃_{k = 1}^{m} (X_{k} - F_{k})) \\ < & \frac{ɛ}{2} . \end{matrix}$

We take $a_{1} = 0, a_{k} = \sum_{i = 1}^{k - 1} p_{i}, k = 2, \dots, m$ , and $p = \sum_{k = 1}^{m} p_{k}$ . For k = 1, 2, ⋯ , m, j= 1, 2, ⋯ , p, we denote $v_{kj} = {\begin{matrix} v_{k (j - β_{k})}^{'}, & if a_{k} < j \leq a_{k + 1}, \\ 0 & otherwise, \end{matrix}$

$θ_{kj} = {\begin{matrix} θ_{k (j - β_{k})}^{'}, & if a_{k} < j \leq a_{k + 1}, \\ 0 & otherwise, \end{matrix}$

$w_{kj} = {\begin{matrix} w_{k (j - β_{k})}^{'}, & if a_{k} < j \leq a_{k + 1}, \\ 0 & otherwise, \end{matrix}$ then, for any k ∈ {1, 2, ⋯ , m}, we have $\sum_{j = 1}^{p} v_{ij} \cdot σ (〈 w_{kj}, x 〉 + θ_{kj}) = \sum_{j = 1}^{p_{k}} v_{ij}^{'} \cdot σ (〈 w_{kj}^{'}, x 〉 + θ_{kj}^{'}) .$

Define $\tilde{H} (x) = \sum_{k = 1}^{m} {\tilde{A}}_{k} \cdot (\sum_{j = 1}^{p} v_{kj} \cdot σ (〈 w_{kj}, x 〉 + θ_{kj})),$ then $\tilde{H} \in H_{0} [σ]$ . Now we prove $𝔇_{Ch} (\tilde{H}, \tilde{Φ}) < ɛ$ .

Noting $D (\tilde{H}, \tilde{Φ}) \leq 1$ , μ (X - F) < ɛ/2, and μ is submodular, by Proposition 2.1, we have $\begin{matrix} 𝔇_{Ch} (\tilde{H}, \tilde{Φ}) \\ = (C) \int_{X} D (\tilde{H}, \tilde{Φ}) d μ \\ = \int_{0}^{\infty} μ ({x : D (\tilde{H}, \tilde{Φ}) \geq t} \cap X) dt \\ = \int_{0}^{1} μ ({x : D (\tilde{H}, \tilde{Φ}) \geq t} \cap X) dt \\ = \int_{0}^{1} μ [({x : D (\tilde{H}, \tilde{Φ}) \geq t} \cap (F \cup (X - F))] dt \\ = \int_{0}^{1} μ [({x : D (\tilde{H}, \tilde{Φ}) \geq t} \cap F) \\ ⋃ ({x : D (\tilde{H}, \tilde{Φ}) \geq t} \cap (X - F))] dt \end{matrix}$

$\begin{matrix} \leq & \int_{0}^{1} μ ([({x : D (\tilde{H}, \tilde{Φ}) \geq t}) \cap F] \\ + [({x : D (\tilde{H}, \tilde{Φ}) \geq t}) \cap (X - F)]) dt \\ \leq & \int_{0}^{1} μ (({x : D (\tilde{H}, \tilde{Φ}) \geq t}) \cap F) dt \\ + \int_{0}^{1} μ ({x : D (\tilde{H}, \tilde{Φ}) \geq t}) \cap (X - F)) dt \\ \leq & \int_{0}^{1} μ (({x : D (\tilde{H}, \tilde{Φ}) \geq t}) \cap F) dt + \frac{ɛ}{2} \end{matrix}$

Now we estimate the first part in the above formula.

Denote $B_{kj} = v_{kj} \cdot σ (〈 w_{kj}, x 〉 + θ_{kj})$ $B_{kj}^{'} = v_{ij}^{'} \cdot σ (〈 w_{kj}^{'}, x 〉 + θ_{kj}^{'}),$ then $\sum_{j = 1}^{p} B_{kj} = \sum_{j = 1}^{p_{k}} B_{kj}^{'}$ .

By using Proposition 4.1, we have $\begin{matrix} μ (({x : D (\tilde{H}, \tilde{Φ}) \geq t}) \cap F) \\ \leq μ ({x : D (\sum_{k = 1}^{m} {\tilde{A}}_{k} \cdot \sum_{j = 1}^{p} B_{kj}, \sum_{k = 1}^{m} χ_{X_{k}} (x) \cdot {\tilde{A}}_{k}) \geq t} \cap F) \\ \leq μ ({x : \sum_{k = 1}^{m} | {\tilde{A}}_{k} | \cdot D (\sum_{j = 1}^{p} B_{kj}, χ_{X_{k}} (x)) \geq t} \cap F) \\ \leq μ ({x : \sum_{k = 1}^{m} | {\tilde{A}}_{k} | \cdot | \sum_{j = 1}^{p} B_{kj} - χ_{X_{k}} (x) | \geq t} \cap F) \\ = μ ({x : \sum_{k = 1}^{m} | {\tilde{A}}_{k} | \cdot | \sum_{j = 1}^{p_{k}} B_{kj}^{'} - χ_{X_{k}} (x) | \geq t} \cap F) \end{matrix}$

On the other hand, if x ∈ F, then for every k = 1, 2, ⋯ , m, we have x ∈ F_k, hence $| χ_{X_{k}} (x) - \sum_{j = 1}^{p_{k}} v_{kj}^{'} \cdot σ (〈 w_{kj}^{'}, x 〉 + θ_{kj}^{'}) | < \frac{ɛ}{2 \sum_{k = 1}^{m} | {\tilde{A}}_{k} |},$ for every k = 1, 2, ⋯ , m. That is, when x ∈ X, $\sum_{k = 1}^{m} | {\tilde{A}}_{k} | \cdot | \sum_{j = 1}^{p_{k}} B_{kj}^{'} - χ_{X_{k}} (x) | < \frac{ɛ}{2} .$ Therefore, $\begin{matrix} \int_{0}^{1} μ (({x : D (\tilde{H}, \tilde{Φ}) \geq t}) \cap F) dt \\ = \int_{0}^{1} μ ({x : \sum_{k = 1}^{m} | {\tilde{A}}_{k} | \cdot | \sum_{j = 1}^{p_{k}} B_{kj}^{'} - χ_{X_{k}} (x) | \geq t} \cap F) dt \\ = \int_{0}^{ɛ / 2} μ ({x : \sum_{k = 1}^{m} | {\tilde{A}}_{k} | \cdot | \sum_{j = 1}^{p_{k}} B_{kj}^{'} - χ_{X_{k}} (x) | \geq t} \cap F) dt \\ < \frac{ɛ}{2} . \end{matrix}$

Thus, we obtain $𝔇_{Ch} (\tilde{H}, \tilde{Φ}) < ɛ$ . The proof of (1) now is complete. □

The following is the limit form of Theorem 4.1.

Theorem 4.2. Let $(ℝ, B)$ be Borel measurable space and μ be a capacity defined on $B$ . If μ is submodular and fulfils the condition (E), then for every fuzzy-value measurable function $\tilde{F} \in L (X)$ , there exist four-layer feedforward RFNNs ${\tilde{H}}_{n} \in H [σ] (n = 1, 2, \dots)$ , such that $lim_{n \to + \infty} (C) \int_{X} D ({\tilde{F}}_{n}, \tilde{F}) d μ = 0,$ i.e., ${\tilde{F}}_{n} \overset{c . m .}{⟶} \tilde{F} (n \to \infty)$ .

From Proposition 2.3, Proposition 2.4, Theorem 4.2, Corollary 3.3 and Corollary 3.4, we can get the following results.

Theorem 4.3. Let $(ℝ, B)$ be Borel measurable space, μ be a submodular capacity and satisfy one of the following conditions:

μ is continuous;

μ is strongly continuous and has property (S) Then the conclusions of Theorems 4.1 and 4.2 hold.

5 Conclusions

We have investigated the approximation property of regular fuzzy neural network with respect to Choquet integral. The Choquet integral is associated with a capacity [14, 15]. When the capacity is submodular [35] and satisfies the condition (E) [29], we get the following result: any fuzzy-valued measurable function can be mean approximated by four-layer regular fuzzy neural network in the sense of Choquet integral.

Observe that at present the machine learning for big data is a hot topic, included also algorithm for big data and intelligent algorithms ([31 , 41] etc.). By applying the results and methods we have obtained to make some researches on these topics is our further work.

Footnotes

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grants No. 11371332 and 11571106).

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