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Self-organizing continuously-overlapping map is shown to have ability to detect the first and second nonlinear principal components. This is an extended version of the self-organizing overlapping mapping. The model was applied to FFT data of sound, and some others. These data are characterized by a combination of two kinds of features, such as the pitch and the quality of tone. The model has two self-organizing layers. One layer extracts and maps continuously one feature, and the other layer does the same with respect to the other feature. The ability of generalization depending on data structure is demonstrated. Comparison to Kohonen's SOM is also discussed.

A neural network for function approximation is treated theoretically. The structure and the information processing of this network is similar to those of the forward-only counterpropagation network, while the learning rule is improved. That is, the learning rule of the hidden layer is self-organizing map instead of winner-take-all. For the case that a target function and the network are with one input and one output, parameters of the learning rule are lead theoretically to obtain the function approximation with the least square error, and the theoretical results are verified by computer simulations.

Dynamic link architecture is self-organizing map applied to image recognition. Mapping formed from data image to sample image. It tends to link two points which have similar local features, and so, can be distorted to some extent. Similarity of two images is evaluated after transforming data image by the mapping. Thus, dynamic link architecture is tolerant to distortion or rotation of data images. We propose a mathematically tractable model of dynamic link architecture, and reduced it to simple phase dynamics, which clearly shows how the model restores rotated data images. Computer simulations verify the theory.

Pattern recognition invariant against deformation or translation can be performed with dynamic link architecture, which has been proposed by von der Malsburg. Dynamic link is applied to many engineering examples efficiently, but has not been analyzed mathematically. We propose a mathematically tractable model of dynamic link architecture. Our model matches common parts between a template image and a data image. To analyze model mathematically, we derive the phase dynamics from the model equation. We also carry out a computer simulation to verify the mathematical theory.