Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)
HOFFMANN, M. and VÁRADY, L., Free-form curve design by neural networks
102 Miklós Hoffmann and Lajos V arady After updating the weights u){j a new input is presented and the next iteration starts. The algorithm determines (using the Euclidean distance) the closest output vector to the presented input vector. The coordinates i.e. the weights of this output vector and those of the vectors that are in a certain neighbourhood of the nearest output vector are updated so that these output vectors get closer to the presented input vector. The degree of the update depends on the gain term and the distance of the output vector and the presented input vector. When the radius (which specifies the neighborhood around an output vector) is large, many output vectors tend towards the presented input. For this reason, initially the output vectors move to places where the density of the input vectors is large, since more input vectors are presented from this areas. The radius (i.e. the size of the neighborhood) and the gain term is decreasing in time. The latter results in that after enough iterations the locations of the output vectors does not change significantly (if the gain term is almost zero then the chänge in the weights is negligible). The gain term should diminish only when the weights are already close to the input vectors. A net is said to be convergent if for all the input vectors P{ (i = 1,..., n) there is an output vector oj such that after a certain time to the EucHdean distance of Oj and Pi is smaller than a predefined limit. A stronger convergence can be obtained if we require that the output vectors which do not converge to an input vector are on the line determined by its two neighbouring output vectors. In the general case the convergence of the Kohonen net has not been proved yet. Kohonen proved the convergence only in a very simple case when the output is one dimensional and the inputs are the elements of an interval (see [2]). The radius, the gain term and the number of the outputs can be adjusted so that the output vectors satisfy the stronger convergence mentioned above. This stronger convergence is important especially in term of the smoothness of the future curve.For the detailed description and evaluation of this problem see [5,6]. Let two converging outputs be o; and Oi + k while the outputs which are between the converging outputs be <?i+i, ..., Oi+k-iThese outputs are in the neighborhoods of the outputs o; and (depending on the radius and k). Since these converged output vectors are close to some input vectors, the outputs o l +\ ,..., will move towards these outputs (and the input vectors). Since they will move to the common line of the ceonverged output vectors. The Kohonen net retains the topological ordering of its output vectors. The weights of two output vectors will be close to each other if the vectors are close on the map. The same is true for the approximated input vectors.
