Approximation of polygonal fuzzy neural networks in sense of Choquet integral norms

Abstract

Approximation capabilities are important and primary properties of neural networks and fuzzy neural networks (FNNs). Neural networks have been successfully applied in many fields since they can work as approximators in nature. Many scholars research FNNs’ approximation abilities for continuous fuzzy functions. It is concluded that FNNs can work as approximators for continuous fuzzy functions if the fuzzy functions satisfy some specified conditions. However, the problem whether FNNs can work as approximators for discontinuous fuzzy functions is not solved completely until now. In this work, we focus on the approximation of polygonal FNN for discontinuous fuzzy functions in sense of Choquet integral norms. We first introduce the Choquet integral norms in sub-additive fuzzy measure. Then the universal approximation of polygonal FNNs for fuzzy valued functions in sense of Choquet integral norms is analyzed in this paper. It is proved that the polygonal FNNs can work as approximators for fuzzy valued functions in the sense of Choquet integral norms with respect to sub-additive fuzzy measure.

Appendix proof of theorems

1.1 Proof of theorem 4.1

Because $\left( {F_{0}^{c} (R),D} \right)$is a complete disjunctive metric space, for arbitrary $\tilde{X} \in F_{0}^{c} (R),i \in N$, there exist an enumerable dense set $\left\{ {\tilde{A}_{i} ,i \in N} \right\}$ such that $D(\tilde{X},\tilde{A}_{i} ) < \varepsilon_{0}$. Let

$$ T_{1} = \left\{ {x \in T|D(F(x),\tilde{A}_{1} ) < \varepsilon } \right\}, $$

$$ T_{2} = \left\{ {x \in T|D(F(x),\tilde{A}_{1} ) \ge \varepsilon ,D(F(x),\tilde{A}_{2} ) < \varepsilon } \right\}, \cdots , $$

$$ \begin{gathered} T_{n} = \{ x \in T|D(F(x),\tilde{A}_{i} ) \ge \varepsilon ,i = 1, \cdots ,n - 1, \hfill \\ D(F(x),\tilde{A}_{n} ) < \varepsilon \} \hfill \\ \end{gathered} $$

where $\mathop \cup \limits_{i \in N} T_{i} = T,T_{i} \cap T_{j} \ne \Upphi$ (i ≠ j) and $\mu (T) = \mu (\mathop \cup \limits_{i \in N} T_{i} ) < + \infty$. Then there exist $d_{0} \in N$ such that $\mu (\mathop \cup \limits_{{i > d_{0} }} T_{i} ) < \varepsilon_{0}$. Let

$$ S\left( x \right) = \sum\limits_{i = 0}^{{d_{0} }} {\left( {a_{i} - a_{i - 1} } \right)} \chi_{{A_{i} (x)}} \left( {x \in T} \right), $$

where $T_{0} = \mathop \cup \limits_{{i > d_{0} }} T_{i} ,a_{0} = 0,0 < a_{1} \le a_{2} \le \cdots \le a_{n} .\{ T_{0} ,T_{1} , \cdots ,T_{{d_{0} }} \}$, is a finite dissection of T, S is the fuzzy-valued simple function on T, and

$$ \begin{gathered} \Updelta (F,S) = (C)\int\limits_{T} {[D (F (x ),S (x ) )]} d\mu \hfill \\ = (C )\int\limits_{{\mathop \cup \limits_{i = 0}^{{d_{0} }} T}} { [D (F (x ) ,S (x ) ) ]} d\mu \hfill \\ \le \sum\limits_{i = 1}^{{d_{0} }} { (C )\int\limits_{{T_{i} }} {D (F (x ) ,S (x ) )} d\mu } \hfill \\ + \int\limits_{{T_{0} }} {D (F_{2} (x )+ F_{3} (x ) )} d\mu \hfill \\ \le \sum\limits_{i = 1}^{{d_{0} }} { (C )\int\limits_{{T_{i} }} {D (F (x ) ,S (x ) )} d\mu } + \varepsilon_{0} \hfill \\ \le \varepsilon_{0} [\mu (T )+ 1 ].\hfill \\ \end{gathered} $$

Suppose $\varepsilon \to 0,{\text{then}}\;\Updelta \left( {F,S} \right) \to 0,$ that is, for arbitrary $\varepsilon > 0,\;\Updelta \left( {F,S} \right) < \varepsilon$ hold. The proof is completed.

1.2 Proof of theorem 4.2

(1) Let

$$ S\left( x \right) = \sum\limits_{k = 1}^{q} {\left( {a_{k} - a_{k - 1} } \right)} \chi_{k} \left( {x \in T} \right) , $$

Where $a_{0} = 0,0 < a_{1} \le a_{2} \le \cdots \le a_{n} .$ For $\chi_{{A_{i} (x)}}$ is a non- negative measurable function, from lemma 4.1, we can conclude that, for arbitrary $\varepsilon_{0} > 0,$ there exist one closed set $L_{k} \subseteq T$ such that

$$ \mu \left( {T - L_{k} } \right) \le \varepsilon_{0} $$

and $\chi_{{A_{i} (x)}}$ is continue on L _k . Therefore there exist a Tauber–Wener function $\sigma ,\,t_{k} \subseteq N,\,v_{k1}^{'} ,v_{k2}^{'} , \cdots ,v_{{kt_{k} }}^{'} ,\,\theta_{k1}^{'} ,\theta_{k2}^{'} , \cdots ,\theta_{{kt_{k} }}^{'} \in R,$ and $u_{k}^{'} (1),u_{k}^{'} (2), \cdots ,\,u_{k}^{'} (t_{k} ) \in R^{n}$ such that

$$ \left| {\chi_{{T_{k} (x)}} - \sum\limits_{j = 1}^{{t_{k} }} {v_{kj}^{'} \cdot \sigma \left( { < u_{k}^{'} (j),x > + \theta_{kj}^{'} } \right)} } \right| < \varepsilon_{0} (x \in L_{k} ). $$

If let $L = \mathop \cap \limits_{k = 1}^{q} L_{k}$, then $T = L \cup (T - L)$, where

$$ \mu \left( {T - L} \right) = \mu \left( {T - \mathop \cap \limits_{k = 1}^{q} L_{k} } \right) = \mu \left( {\mathop \cup \limits_{k = 1}^{q} (T - L_{k} )} \right) \le \sum\limits_{k = 1}^{q} {\mu \left( {T - L_{k} } \right)} . $$

Because $\mu \left( {T - L_{k} } \right) < \varepsilon_{0}$, there exist ɛ ₁ > 0 such that $\mu \left( {T - L} \right) < \varepsilon_{1}$. Suppose $t = \sum\limits_{k = 1}^{q} {t_{k} }$, denote $\beta_{k} = \sum\nolimits_{r = 1}^{k - 1} {t_{r} } ,\beta_{1} = 0$ For $k = 1,2, \cdots ,q,j = 1,2, \cdots ,t,$ define

$$ v_{kj} = \left\{ {\begin{array}{*{20}c} {v'_{{k(j - \beta_{k} )}} ,\beta_{k} < j \le \beta_{k + 1} ,} \\ {0,\;\;\;{\text{otherwise}}.} \\ \end{array} } \right. $$

$$ u_{k} (j) = \left\{ {\begin{array}{*{20}l} u'_{k} (j - \beta_{k} ), & \beta_{k} < j \le \beta_{k + 1} , \\ 0 & {\text{otherwise}}. \end{array} } \right. $$

$$ \theta_{kj} = \left\{ {\begin{array}{*{20}l} \theta'_{k(j - \beta_{k})}, & \beta_{k}<j \le \beta_{k + 1}, \\ 0 &{\text{otherwise}} \end{array}}\right. $$

Let

$$ H\left( x \right) = \sum\limits_{k = 1}^{q} {\tilde{A}_{k} } \cdot \sum\limits_{j = 1}^{t} {v_{kj} \cdot \sigma \left( { < u_{k} (j),x > + \theta_{kj} } \right)} , $$

Then $H \in H_{0} [\sigma ]$ is the fuzzy measure space and for $\forall k = \left\{ {1,2, \cdots ,q} \right\},$

$$ \begin{gathered} \sum\limits_{j = 1}^{t} {v_{kj} \cdot \sigma \left( { < u_{k} (j),x > + \theta_{kj} } \right)} \hfill \\ \; = \sum\limits_{j = 1}^{t} {v'_{kj} \cdot \sigma \left( { < u'_{k} (j),x > + \theta '_{kj} } \right).} \hfill \\ \end{gathered} $$

$$ \begin{gathered} \Updelta (H,S) = (C)\int\limits_{T} {D(H(x),S(x))d\mu } = (C)\int\limits_{L \cup (T - L)} {D(H(x),S(x))} d\mu \hfill \\ \le (C)\int\limits_{L \cup (T - L)} {D\left( {\sum\limits_{k = 1}^{q} {\tilde{A}_{k} \cdot \sum\limits_{j = 1}^{t} {v_{kj} \cdot \sigma \left( { < u_{k} (j),x > + \theta_{kj} } \right)} ,\sum\limits_{k = 1}^{q} {(a_{k} - a_{k - 1} )\chi_{k} } } } \right)} d\mu \hfill \\ \le \sum\limits_{k = 1}^{q} {(\tilde{A}_{k} \vee \left| {a_{k} - a_{k - 1} } \right|) \cdot } (C)\int\limits_{L \cup (T - L)} {D\left( {\sum\limits_{j = 1}^{t} {v_{kj} \cdot \sigma \left( { < u_{k} (j),x > + \theta_{kj} } \right),\chi_{k} } } \right)} d\mu \hfill \\ \le \sum\limits_{k = 1}^{q} {(\tilde{A}_{k} \vee \left| {a_{k} - a_{k - 1} } \right|) \cdot \left( {(C)\int\limits_{L} {D\left( {\sum\limits_{j = 1}^{t} {v_{kj} \cdot \sigma \left( { < u_{k} (j),x > + \theta_{kj} } \right),\chi_{k} } } \right)} d\mu + } \right.} \hfill \\ \left. {\;(C)\int\limits_{(T - L)} {D\left( {\sum\limits_{j = 1}^{t} {v_{kj} \cdot \sigma \left( { < u_{k} (j),x > + \theta_{kj} } \right),\chi_{k} } } \right)} d\mu } \right) \hfill \\ \le \sum\limits_{k = 1}^{q} {(\tilde{A}_{k} \vee \left| {a_{k} - a_{k - 1} } \right|) \cdot } \left( {(C)\int\limits_{L} {\varepsilon_{0} } d\mu } \right. + \hfill \\ \;\,\;\left. {(C)\int\limits_{(T - L)} {D\left( {\sum\limits_{j = 1}^{t} {v_{kj} \cdot \sigma \left( { < u_{k} (j),x > + \theta_{kj} } \right),\chi_{k} } } \right)} d\mu } \right) \hfill \\ \end{gathered} $$

For $\mu \left( {T - L} \right) < \varepsilon_{1}$, there exist ɛ ₂ > 0 such that $(C)\int_{T - L} {\left[ {D\left( {\sum\limits_{j = 1}^{t} {v_{kj} \cdot \sigma \left( { < u_{k} (j) + \theta_{kj} > } \right),\chi_{k} } } \right)} \right]d\mu < \varepsilon_{2} }$. Therefore

$$ \begin{gathered} \Updelta (H,S) \le \sum\limits_{k = 1}^{q} {(\tilde{A}_{k} \vee \left| {a_{k} - a_{k - 1} } \right|) \cdot \left( {(C)\int\limits_{L} {\varepsilon_{0} } d\mu + } \right.} \hfill \\ \left. {\;\;\;\;(C)\int\limits_{(T - L)} {D\left( {\sum\limits_{j = 1}^{t} {v_{kj} \cdot \sigma \left( { < u_{k} (j),x > + \theta_{kj} } \right),\chi_{k} } } \right)} d\mu } \right) \hfill \\ \; \le \sum\limits_{k = 1}^{q} {(\tilde{A}_{k} \vee \left| {a_{k} - a_{k - 1} } \right|) \cdot } \left( {(C)\int\limits_{L} {\varepsilon_{0} } d\mu + \varepsilon_{2} } \right) \hfill \\ \; \le \sum\limits_{k = 1}^{q} {(\tilde{A}_{k} \vee \left| {a_{k} - a_{k - 1} } \right|) \cdot } \left( {\varepsilon_{0} \cdot \mu (L) + \varepsilon_{2} } \right) \hfill \\ \end{gathered} $$

That is to say, there exist ɛ > 0 such that $\Updelta \left( {H,S} \right) < \varepsilon$. The proof of (1) is completed.

(2) We can conclude (2) from theorem 4.1 and (1).