Stark's Neural Network Algorithm

Stark's algorithm [3,4] implies the transition from a given IFS F to the corresponding neural network W. To do such a passage, it is enough to consider any one iteration of F.

Let $X \equiv S^u \subset {\rm I\!R}^2$, S^u - a unit square digitized with respect to fixed resolutions $\Delta_x$ and $\Delta_y$. Every A_n has its indicator function, $y: S^u \rightarrow \{0,1\}$, such that:

$$\forall s \in S^u: \;\; y_s(n)= \left\{ \begin{array}{ll} 1... ..._n$ } \\ 0 & \mbox{if $s \not\in A_n$ } \end{array} \right.$$ (8)

If we define

$$w_{ss\prime}= \left\{ \begin{array}{lll} 1 & \mbox{if $f_i(... ...for some $i$ } \\ 0 & {\rm otherwise} & \end{array} \right.$$ (9)

then the dynamics of y_s(n) is given by

$$y_s(n+1)=g\Bigl(\sum_{s\prime \in S}w_{ss\prime}y_{s\prime}(n)\Bigr),$$ (10)

where g is a step function.

Thus, we have the binary neural network with |S^u| neurons y_s and synaptic weights $w_{ss\prime}$ that realizes the IFS F.