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Approximation of fuzzy neural networks

by using Lusin’s theorem

Jun Li; Commun. Univ. of China, China Jianzeng Li; Commun. Univ. of China, China

Masami Yasuda; Chiba University, Japan

September 23, 2006

Abstract

Abstract Introduction Preliminaries Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem

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In this note, we study an approximation property of regular fuzzy neural network(RFNN). It is shown that any fuzzy-valued measurable function can be approximated by the four-layer RFNN in the sense of fuzzy integral norm for the finite sub-additive fuzzy measure on ℝ.

Keywords: Fuzzy measure; Lusin’s theorem; Approximation; Regular fuzzy neural network

Introduction

Abstract Introduction Introduction 1/4 Introduction 2/4 Introduction 3/4 Introduction 4/4 Preliminaries Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem

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Introduction 1/4

Abstract Introduction Introduction 1/4 Introduction 2/4 Introduction 3/4 Introduction 4/4 Preliminaries Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem

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In neural network theory, 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.

Abstract Introduction Introduction 1/4 Introduction 2/4 Introduction 3/4 Introduction 4/4 Preliminaries Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem

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The studies on this matter were undertaken by many authors and a great number of important results were obtained.

• J.G Attali, G. Pages, Approximation of functions by a multilayer perceptron: a new approach, Neural Networks 10(1997) 1069-1081,

• R.M Burton, H.G. Dehling, Universal approximation in p-mean by neural networks, Neural Networks 11(1998) 661-667,

• F Scarselli, A.G. Tsoi, Universal approximation using feedforward neural networks: a survey of some existing methods, and some new results, Neural Networks 11(1998) 15-17.

Abstract Introduction Introduction 1/4 Introduction 2/4 Introduction 3/4 Introduction 4/4 Preliminaries Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem

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The similar approximation problems in fuzzy environment were investigated by

• J.J Buckley, Y. Hayashi, Fuzzy input-output controllers are universal approximators, Fuzzy Sets and Systems 58(1993) 273-278,

• J.J Buckley, Y. Hayashi, Can fuzzy neural nets approximate continuous fuzzy function, Fuzzy Sets and Systems 61(1994) 43-51,

• P. Liu, Analyses of regular fuzzy neural networks for approximation capabilities, Fuzzy Sets and Systems 114(2000) 329-338,

• P. Liu, Universal approximations of continuous fuzzy-valued functions by multi-layer regular fuzzy neural networks, Fuzzy Sets and Systems 119(2001) 313-320.

Abstract Introduction Introduction 1/4 Introduction 2/4 Introduction 3/4 Introduction 4/4 Preliminaries Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem

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In P. Liu(2001) proved that 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 this note, by using Lusin’s theorem on fuzzy measure space, we show that such RFNNs is pan-approximator for fuzzy-valued measurable function. That is, any fuzzy-valued measurable function can be approximated by the four-layer RFNNs in the sense of fuzzy integral norm for the finite sub-additive measure on ℝ.

Preliminaries

Abstract Introduction Preliminaries Preliminaries 1/3 Preliminaries 2/3 Preliminaries 3/3 Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem

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Preliminaries 1/3

Abstract Introduction Preliminaries Preliminaries 1/3 Preliminaries 2/3 Preliminaries 3/3 Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem

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We suppose that (X, ρ) is a metric space, and that O and C are the classes of all open and closed sets in (X, ρ), respectively, and B is Borel σ-algebra on X, i.e., it is the smallest σ-algebra containing O. A set function μ : B → [0, +∞) is called a fuzzy measure(Narukawa/Murofushi(2004)), if it satisfies the following properties:

(FM1) μ(∅) = 0; (FM2) A ⊂ B implies μ(A) ≤ μ(B).

A fuzzy measure μ is called null-additive (Wang/Klir(1992)), if for any E, F ∈ B and μ(F) = 0 imply μ(E ∪ F) = μ(E); sub-additive (Pap(1995)), if for any E, F ∈ B we have μ(E ∪ F) ≤ μ(E) + μ(F).

Abstract Introduction Preliminaries Preliminaries 1/3 Preliminaries 2/3 Preliminaries 3/3 Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem

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In this paper, we always assume that μ is a finite, sub-additive and continuous fuzzy measure on B. Consider a nonnegative real-valued measurable function f on A and the fuzzy integral of f on A with respect to μ, which is denoted by

\begin{aligned}& (S) ∫_{A} f dμ \\ ≜ & sup_{0≤α<+∞} [α ∧ μ(\{x : f(x) ≥ α\} ∩ A)] \end{aligned}

Abstract Introduction Preliminaries Preliminaries 1/3 Preliminaries 2/3 Preliminaries 3/3 Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem

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Theorem 0.1 (Lusin’s theorem (Li/Yasuda(2004), Song/Li(2003)) Let (X, ρ) be metric space and μ be null additive fuzzy measure on B. If f is a real-valued measurable function on E ∈ B, then, for every ϵ > 0, there exists a closed subset F_{ϵ} ∈ B such that f is continuous on F_{ϵ} and μ(E − F_{ϵ}) < ϵ.

Approximation in fuzzy mean by RFNN

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10

Main Theorem

Pf of Main Theorem

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Approximation 1/10

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10

Main Theorem

Pf of Main Theorem

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In this section, we study an approximation property of the four-layer RFNNs to fuzzy-valued measurable function in the sense of fuzzy integral norm for fuzzy measure on ℝ.

Abstract Introduction Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10 Main Theorem Pf of Main Theorem

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Let F_{0}(ℝ) be the set of all bounded fuzzy numbers, i.e., for Ã ∈ F_{0}(ℝ), the following conditions hold:

(i) ∀ α ∈ (0, 1], Ã_{α} ≜ {x ∈ ℝ | Ã(x) ≥ α} is the closed interval of ℝ; (ii) The support Supp(Ã) ≜ cl{x ∈ ℝ | Ã(x) > 0} ⊂ is a bounded set; (iii) {x ∈ ℝ | Ã(x) = 1} ≠ ∅.

For simplicity, supp(Ã) is also written as Ã_{0}. Obviously, Ã_{0} is a bounded and closed interval of ℝ. For Ã ∈ F_{0}(ℝ), let Ã_{α} = [a_{α}^{−}, a_{α}^{+}] for each α ∈ [0, 1] and we denote

|Ã| ≜ ⋁_{α∈[0,1]}(|a_{α}^{−}| ∨ |a_{α}^{+}|).

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10

Main Theorem

Pf of Main Theorem

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Proposition 0.1 Liu(2001) Assume Ã, Ã_{1}, Ã_{2} ∈ F_{0}(ℝ), and W̃_{i}, Ṽ_{i} ∈ F_{0}(ℝ)(i = 1, 2, ⋯, n). Then

(1) d(Ã · Ã_{1}, Ã · Ã_{2}) ≤ |Ã| · d(Ã_{1}, Ã_{2}),

(2) d(∑_{i=1}^{n} W̃_{i}, ∑_{i=1}^{n} Ṽ_{i}) ≤ ∑_{i=1}^{n} d(W̃_{i}, Ṽ_{i}).

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10

Main Theorem

Pf of Main Theorem

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For Ã, B̃ ∈ F_{0}(ℝ), define metric d(Ã, B̃) between Ã and B̃ by

d(Ã, B̃) ≜ sup_{α∈[0,1]} d_{H}(Ã_{α}, B̃_{α})

where d_{H} means Hausdorff metric: for A, B ⊂ ℝ,

d_{H}(A, B)

≜ max {sup_{x∈A} inf_{y∈B}(|x − y|), sup_{y∈B} inf_{x∈A}(|x − y|)} .

Abstract Introduction Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10 Main Theorem Pf of Main Theorem

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It is known that (F_{0}(ℝ), d) is a completely separable metric space (Diamond/Kloeden(1994)). Let T be a measurable set in ℝ^{n}, (T, B ∩ T, μ) finite fuzzy measure space. Let L(T) denote the set of all fuzzy-valued measurable function

F̃ : T → F_{0}(ℝ).

For any F̃_{1}, F̃_{2} ∈ L(T), d(F̃_{1}, F̃_{2}) is measurable function on (T, B ∩ T), we will write a fuzzy integral norm as

△_{S}(F̃_{1}, F̃_{2}) ≜ (S) ∫_{T} d(F̃_{1}, F̃_{2})dμ.

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10

Main Theorem

Pf of Main Theorem

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Proposition 0.2 Let F̃_{1}, F̃_{2}, F̃_{3} ∈ L(T), then

△_{S}(F̃_{1}, F̃_{3}) ≤ 2(△_{S}(F̃_{1}, F̃_{2}) + △_{S}(F̃_{2}, F̃_{3})).

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10

Main Theorem

Pf of Main Theorem

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Proof. From subadditivity of μ, we have

△_{S}(F̃_{1}, F̃_{3}) = (S) ∫_{T} d(F̃_{1}, F̃_{3})dμ

= ⋁_{α∈[0,∞)}{α ∧ μ(T ∩ (d(F̃_{1}, F̃_{3}))_{α}}

≤ ⋁_{α∈[0,∞)}{α ∧ μ(T ∩

(d(F̃_{1}, F̃_{2})\frac{α}{2} ∪ d(F̃_{2}, F̃_{3})\frac{α}{2} ))\}

≤ ⋁_{α∈[0,∞)}\{α ∧ [μ(T ∩ d(F̃_{1}, F̃_{2})\frac{α}{2} )

+μ(T ∩ d(F̃_{2}, F̃_{3})\frac{α}{2} )]\}.

Because of the elementary inequality: a ∧ (b + c) ≤ (a ∧ b) + (a ∧ c) where a, b, c ≥ 0,

Approximation 8a/10

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10

Main Theorem

Pf of Main Theorem

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\begin{aligned}we have \\ & △_{S}(F̃_{1}, F̃_{3}) & ≤ & ⋁_{α∈[0,∞)}\{α ∧ μ(T ∩ d(F̃_{1}, F̃_{2})\frac{α}{2} ) \\ & & & +α ∧ μ(T ∩ d(F̃_{2}, F̃_{3})\frac{α}{2} )\} \\ & & ≤ & ⋁_{α∈[0,∞)}[α ∧ μ(T ∩ d(F̃_{1}, F̃_{2})\frac{α}{2} )] \\ & & & + ⋁_{α∈[0,∞)}[α ∧ μ(T ∩ d(F̃_{2}, F̃_{3})\frac{α}{2} )] \end{aligned}

Approximation 8b/10

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10

Main Theorem

Pf of Main Theorem

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\begin{aligned}≤ & ⋁_{α∈[0,∞)} [\frac{α}{2} ∧ μ(T ∩ d(F̃_{1}, F̃_{2})\frac{α}{2} ) \\ & + \frac{α}{2} ∧ μ(T ∩ d(F̃_{1}, F̃_{2})\frac{α}{2} )] \\ & + ⋁_{α∈[0,∞)} [\frac{α}{2} ∧ μ(T ∩ d(F̃_{2}, F̃_{3})\frac{α}{2} ) \\ & + \frac{α}{2} ∧ μ(T ∩ d(F̃_{2}, F̃_{3})\frac{α}{2} )] \end{aligned}

Approximation 8c/10

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10

Main Theorem

Pf of Main Theorem

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\begin{aligned}≤ & ⋁_{\frac{α}{2} ∈[0,∞)} [\frac{α}{2} ∧ μ(T ∩ d(F̃_{1}, F̃_{2})\frac{α}{2} )] \\ & + ⋁_{\frac{α}{2} ∈[0,∞)} [\frac{α}{2} ∧ μ(T ∩ d(F̃_{1}, F̃_{2})\frac{α}{2} )] \\ & + ⋁_{\frac{α}{2} ∈[0,∞)} [\frac{α}{2} ∧ μ(T ∩ d(F̃_{2}, F̃_{3})\frac{α}{2} )] \\ & + ⋁_{\frac{α}{2} ∈[0,∞)} [\frac{α}{2} ∧ μ(T ∩ d(F̃_{2}, F̃_{3})\frac{α}{2} )] \\ = & 2 (△_{S}(F̃_{1}, F̃_{2}) + △_{S}(F̃_{2}, F̃_{3})) . \end{aligned}

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10

Main Theorem

Pf of Main Theorem

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Definition 0.1 (Liu(2001)) A fuzzy-valued function Φ̃ : T → F_{0}(ℝ) is called a fuzzy-valued simple function, if there exist Ã_{1}, Ã_{2}, …, Ã_{m} ∈ F_{0}(ℝ), such that ∀ x ∈ T,

Φ̃(x) = ∑_{k=1}^{m} Ã_{k} · χ_{T_{k}}(x)

where T_{k} ∈ B ∩ T (k = 1, 2, …, m), T_{i} ∩ T_{j} = ∅ (i ≠ j) and

T = ⋃_{k=1}^{m} T_{k}.

Abstract Introduction Preliminaries Approximation in fuzzy mean by RFNN Approximation 1/10 Approximation 2/10 Approximation 3/10 Approximation 4/10 Approximation 5/10 Approximation 6/10 Approximation 7/10 Approximation 8a/10 Approximation 8b/10 Approximation 8c/10 Approximation 9/10 Approximation 10/10 Main Theorem Pf of Main Theorem

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Immediately, if S(T) denotes the set of all fuzzy-valued simple functions, then S(T) ⊂ L(T). Similar to the proof of Proposition 0.2 and by using subadditivity of μ, we can obtain the following proposition.

Proposition 0.3 Let μ be a finite, sub-additive and continuous fuzzy measure on ℝ. If F̃ ∈ L(T), then for every ϵ > 0, there exists Φ̃_{ϵ} ∈ S(T) such that

△_{S}(F̃, Φ̃_{ϵ}) < ϵ.

Main Theorem

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem Main Theorem 1/6 Main Theorem 2/6 Main Theorem 3/6 Main Theorem 4/6 Main Theorem 5/6 Main Theorem 6/6

Pf of Main Theorem

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Main Theorem 1/6

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem Main Theorem 1/6 Main Theorem 2/6 Main Theorem 3/6 Main Theorem 4/6 Main Theorem 5/6 Main Theorem 6/6

Pf of Main Theorem

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Define

H[σ] ≜ {H̃ | H̃(x) = ∑_{i=1}^{n} W̃_{i} V_{i}[σ]}

where

V_{i}[σ] ≜ ∑_{j=1}^{m} Ṽ_{ij} · σ(x · Ũ_{j} + Θ̃_{j})

and σ is a given extended function of σ : ℝ → ℝ (bounded, continuous and nonconstant), and x ∈ ℝ, W̃_{i}, Ṽ_{ij}, Ũ_{j}, Θ̃_{j} ∈ F_{0}(ℝ).

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem Main Theorem 1/6 Main Theorem 2/6 Main Theorem 3/6 Main Theorem 4/6 Main Theorem 5/6 Main Theorem 6/6

Pf of Main Theorem

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For any H̃ ∈ H[σ], H̃ is a four-layer feedforward RFNN with activation function σ, threshold vector (Θ̃_{1}, …, Θ̃_{m}) in the first hidden layer(cf. Liu(2001).

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem Main Theorem 1/6 Main Theorem 2/6 Main Theorem 3/6 Main Theorem 4/6 Main Theorem 5/6 Main Theorem 6/6

Pf of Main Theorem

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Restricting fuzzy numbers Ṽ_{ij}, Ũ_{j}, Θ̃_{j} ∈ F_{0}(ℝ), respectively, to be real

numbers v_{ij}, u_{j}, θ_{j} ∈ ℝ, we obtain the subset H_{0}[σ] of H[σ]:

H_{0}[σ] ≜ {H̃ | H̃(x) = ∑_{i=1}^{n} W̃_{i} v_{i}[σ]} .

where

v_{i}[σ] ≜ ∑_{j=1}^{m} v_{ij} · σ(x · u_{j} + θ_{j}).

Abstract Introduction Preliminaries Approximation in fuzzy mean by RFNN Main Theorem Main Theorem 1/6 Main Theorem 2/6 Main Theorem 3/6 Main Theorem 4/6 Main Theorem 5/6 Main Theorem 6/6 Pf of Main Theorem

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Let define two classes of pan-approximation which is fundamental to our results.

Definition 0.2

(1) H_{0}[σ] is call the pan-approximator of S(T) in the sense of △_{S}, if for ∀ Φ̃ ∈ S(T), ∀ ϵ > 0, there exists H̃_{ϵ} ∈ H_{0}[σ] such that △_{S}(Φ̃, H̃_{ϵ}) < ϵ.

(2) For F̃ ∈ L(T), H[σ] is call the pan-approximator for F̃ in the sense of △_{S}, if ∀ ϵ > 0, there exists H̃_{ϵ} ∈ H[σ] such that △_{S}(F̃, H̃_{ϵ}) < ϵ.

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem Main Theorem 1/6 Main Theorem 2/6 Main Theorem 3/6 Main Theorem 4/6 Main Theorem 5/6 Main Theorem 6/6

Pf of Main Theorem

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By using Lusin’s theorem (Theorem 0.1), Proposition 0.2 and 0.3 we can obtain the main result in this paper, which is stated in the following.

Abstract Introduction Preliminaries Approximation in fuzzy mean by RFNN Main Theorem Main Theorem 1/6 Main Theorem 2/6 Main Theorem 3/6 Main Theorem 4/6 Main Theorem 5/6 Main Theorem 6/6 Pf of Main Theorem

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Theorem 0.2 Let (T, B ∩ T, μ) be fuzzy measure space and μ be finite, sub-additive and continuous. Then,

(1) H_{0}[σ] is the pan-approximator of S(T) in the sense of △_{S}.

(2) H[σ] is the pan-approximator for F̃ in the sense of △_{S}.

Pf of Main Theorem

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Pf of Main Thm 1/13

Abstract Introduction Preliminaries Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Proof. By using the conclusion of (1) and Proposition 0.3 we can obtain (2). Now we only prove (1). Suppose that Φ̃(x) is a fuzzy-valued simple function, i.e.,

Φ̃(x) = ∑_{k=1}^{m} χ_{T_{k}}(x) · Ã_{k} (x ∈ T).

For arbitrarily given ϵ > 0, applying Theorem 0.1 (Lusin’s theorem) to each real measurable function χ_{T_{k}}(x), for every fixed k (1 ≤ k ≤ m), there exists closed set F_{k} ∈ B ∩ T such that

F_{k} ⊂ L_{k} and μ(L_{k} − F_{k}) < \frac{ϵ}{2m}

and χ_{T_{k}}(x) is continuous on F_{k}.

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Therefore, for every k there exist a Tauber-Wiener function σ and

p_{k} ∈ N, vk1^{′}, vk2^{′}, ⋯, vkp_{k}^{′}, θk1^{′}, θk2^{′}, ⋯, θkp_{k}^{′} ∈ ℝ, and

wk1^{′}, wk2^{′}, ⋯, wkp_{k}^{′} ∈ ℝ^{n} such that

| χ_{T_{k}}(x) − ∑_{j=1}^{p_{k}} vkj^{′} · σ(⟨wkj^{′}, x⟩ + θkj^{′}) |< \frac{ϵ}{2 ∑_{k=1}^{m} |Ã_{k}|}

for x ∈ L_{k}. Note that we can assume ∑_{k=1}^{m} |Ã_{k}| ≠ 0, without any loss of generality.

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Denote L = ⋂_{k=1}^{m} L_{k}, then T = L ∪ (T − L). By the subadditivity of

μ, we have

\begin{aligned}μ(T − L) & = μ(⋃_{k=1}^{m}(T − L_{k})) \\ & ≤ ∑_{k=1}^{m} μ(T − L_{k}) < \frac{ϵ}{2}. \end{aligned}

We take β_{1} = 0, β_{k} = ∑_{i=1}^{k−1} p_{i}, k = 2, ⋯, m, and p = ∑_{k=1}^{m} p_{k}. For

k = 1, 2, ⋯, m, j = 1, 2, ⋯, p, we denote

v_{kj} = \{ \begin{matrix}vk(j−β_{k})^{′}, & if β_{k} < j ≤ β_{k+1}, \\ 0 & otherwise, \end{matrix}

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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θ_{kj} = \{ \begin{matrix}θk(j−β_{k})^{′}, & if β_{k} < j ≤ β_{k+1}, \\ 0 & otherwise, \end{matrix}

w_{kj} = \{ \begin{matrix}wk(j−β_{k})^{′}, & if β_{k} < j ≤ β_{k+1}, \\ 0 & otherwise, \end{matrix}

then, for any k ∈ {1, 2, ⋯, m}, we have

\begin{matrix}∑_{j=1}^{p} v_{ij} · σ(⟨w_{kj}, x⟩ + θ_{kj}) \\ = ∑_{j=1}^{p_{k}} vij^{′} · σ(⟨wkj^{′}, x⟩ + θkj^{′}). \end{matrix}

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Now denote that

H̃(x) = ∑_{k=1}^{m} Ã_{k} · (∑_{j=1}^{p} v_{kj} · σ(⟨w_{kj}, x⟩ + θ_{kj})) ,

then H̃ ∈ H_{0}[σ].

In the reminder part of this section we will prove △_{S}(H̃, Φ̃) < ϵ.

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Denote

B_{kj} = v_{kj} · σ(⟨w_{kj}, x⟩ + θ_{kj})

and

Bkj^{′} = vij^{′} · σ(⟨wkj^{′}, x⟩ + θkj^{′}).

By using Proposition 0.1 and noting μ(T − L) < ϵ/2, we have

△_{S}(H̃, Φ̃)

= (S) ∫_{T} d(H̃, Φ̃)dμ

= ⋁_{0≤α<+∞} [α ∧ μ(T ∩ (d(H̃, Φ̃))_{α})] .

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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\begin{aligned}Since \\ & & μ(T ∩ d(H̃, Φ̃)_{α}) \\ & = & μ (T ∩ d (∑_{k=1}^{m} Ã_{k} · ∑_{j=1}^{p} B_{kj}, \\ ∑_{k=1}^{m} χ_{T_{k}}(x) · Ã_{k}) & α & ) \\ & ≤ & μ ((L ∪ (T − L)) ∩ (C_{mp})_{α}) \\ & ≤ & μ (L ∩ (C_{mp})_{α}) + μ ((T − L) ∩ (C_{mp})_{α}) \end{aligned}

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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where the notation of set C_{mp} is assigned as

C_{mp} = ∑_{k=1}^{m} |Ã_{k}| · d (∑_{k=1}^{p} B_{kj}, χ_{T_{k}}(x)) .

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Hence △_{S}(H̃, Φ̃) is dominated by

⋁_{0≤α<+∞} [α ∧ μ (L ∩ (D_{mp})_{α})]

+ ⋁_{0≤α<+∞} [α ∧ μ ((T − L))]

≤ ⋁_{0≤α<+∞} [α ∧ μ (L ∩ (D′_{mp})_{α})] + \frac{ϵ}{2}

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Here, for simplicity, we use the notations of

D_{mp} = ∑_{k=1}^{m} |Ã_{k}| · |∑_{j=1}^{p} B_{kj} − χ_{T_{k}}(x)|

and

D′_{mp} = ∑_{k=1}^{m} |Ã_{k}| · d (∑_{j=1}^{p_{k}} Bkj^{′} − χ_{T_{k}}(x)) .

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Now we estimate the first part in the above formula. If x ∈ L, then for every k = 1, 2, ⋯, m, we have x ∈ L_{k}, hence

| χ_{T_{k}}(x) − ∑_{j=1}^{p_{k}} vkj^{′} · σ(⟨wkj^{′}, x⟩ + θkj^{′}) |< \frac{ϵ}{2 ∑_{k=1}^{m} |Ã_{k}|},

for every k = 1, 2, ⋯, m. That is, for x ∈ L,

D′_{mp} = ∑_{k=1}^{m} |Ã_{k}| · d (∑_{j=1}^{p_{k}} Bkj^{′} − χ_{T_{k}}(x)) < \frac{ϵ}{2}.

Abstract

Introduction

Preliminaries Approximation in fuzzy mean by RFNN

Main Theorem

Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Therefore,

⋁_{0≤α<+∞} [α ∧ μ (L ∩ (D′_{mp})_{α})]

= ⋁_{α∈[0, \frac{ϵ}{2} ]} [α ∧ μ (L ∩ (D′_{mp})_{α})]

+ ⋁_{α∈[ \frac{ϵ}{2} ,∞)} [α ∧ μ (L ∩ (D′_{mp})_{α})]

= ⋁_{α∈[0, \frac{ϵ}{2} ]} [α ∧ μ (L ∩ (D′_{mp})_{α})]

≤ \frac{ϵ}{2}.

Abstract

Introduction

Preliminaries

Approximation in fuzzy mean by RFNN Main Theorem Pf of Main Theorem Pf of Main Thm 1/13 Pf of Main Thm 2/13 Pf of Main Thm 3/13 Pf of Main Thm 4/13 Pf of Main Thm 5/13 Pf of Main Thm 6/13 Pf of Main Thm 7/13 Pf of Main Thm 8/13 Pf of Main Thm 9/13 Pf of Main Thm 10/13 Pf of Main Thm 11/13 Pf of Main Thm 12/13 Pf of Main Thm 13/13

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Thus, combining with the previous evaluation, we obtain

△_{S}(H̃, Φ̃)
≤ ⋁_{0≤α<+∞} [α ∧ μ (L ∩ (D′_{mp})_{α})] + \frac{ϵ}{2}
< ϵ.

The proof of (1) is now completed. □

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