J. G oppert and W. Rosenstiel, "Topology preserving interpolation in self-organizing maps," in Proc. NEURONIMES '93, 1993, pp. 425--434.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....where #(t) and h j,v(x) designate the learning rate and the neighbourhood function centered around the winner, respectively. If one deals with continuous signals, an interpolation between the winner and its neighbours is then necessary in order to make the map continuous. Geometric interpolation [9] uses orthogonal projections of the vector formed by the approximation to the exact input onto the one formed by the approximation to the second best matching unit. Topological interpolation [10] instead, is based on a selection of the topological neighbours of the winner, which is an advantage ....

J. G oppert and W. Rosenstiel, "Topology preserving interpolation in self-organizing maps," in Proc. NEURONIMES '93, 1993, pp. 425--434.

....of Vorono regions, this model drags with it the drawbacks of the RBF networks described above and also the problem of the symmetrical shape of the Gaussians which leads to influence equally the neighboring kernels of the input space without any consideration for the real topology of the map. In [9], an interpolation method is proposed using a distance measure between the input point and the closest (Euclidean distance) neurons of a SOM. Although it respects the topology of the map and provides good results compared to other function approximation techniques, it does not provide a first ....

J.Gppert and W.Rosenstiel, Topology-preserving interpolation in self-organizing maps, NeuroNimes '93, 1993, 425-434.

....a local framework liable to limit the interference phenomenon and preserving the topology of the data using neighborhood links between the neurons. SOMs have been used in [2] 9] 10] 11] with a first order expansion around each neuron in the output space, called Local Linear Mapping (LLM) and in [12] with a topological interpolation technique in order to solve function approximation problems. But the presence of discontinuities of the output function for input vectors lying along the border of Vorono regions in such approaches put a strain on their approximation quality. We have proposed ....

J.Gppert & W.Rosenstiel, Topology-preserving interpolation in self-organizing maps, NeuroNimes'93, 425-434, 1993.

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J. G oppert and W. Rosenstiel. Topology-preserving interpolation in self-organizing maps. In Proceedings of NeuroNimes 93, pages 425--434, Nanterre, France, 10 1993. EC2.

....data (196 components) we use the self organizing map in order to reduce the dimensionality. For the classification of the topological structure of the error an output layer has been added. Instead of using the standard counterpropagation [HN89] we have used the geometrical interpolation again ( G o93] and [Lud95] The results of this algorithm have been compared with results of counterpropagation (CPN) topological interpolation in the outstar layer of a counterpropagation net [G o92] and resilient propagation (Rprop) Bra93] 3.2 SOM for the analysis of PCM data For the problem of ....

J. G oppert and W. Rosenstiel. Topology-preserving interpolation in self-organizing maps. In Proceedings of NeuroNimes 93, pages 425--434, Nanterre, France, 10 1993. EC2.

....vectors will be represented by neighbouring units, but only, if they have been trained in a common context. Dynamic training can be seen as an extension from topological feature space into spatiotemporal space of continuous varying stimuli. A combination of this map with interpolation principles [Gop93a] may lead to a completely new representation of continuous spatio temporal data in a continuous ordered map. Context leads also to new principles for the recognition of ambiguous data. If the same stimulus occurs in several ways and in different contexts, the use of some neighbouring inputs may ....

J. Goppert and W. Rosenstiel. Topology-preserving interpolation in self-organizing maps. In Proc. of NeuroNimes 93, Nanterre, France, 1993. EC2.

....But a serious problem occurs: The number of neurons and consequently the calculation time needed for finding the winner, as well as the memory for storing the weights increases exponentially. This effect are drastically reduced by applying local linearization and interpolation. In previous works [G op93] different methods, based on the projection of error vectors onto the distance vector to further winners have been presented. Different applications have shown the aptitude of these methods and the properties compared to other neural network techniques. A method with a similar aim, but another ....

....codebooks (W (in) in a topologically correct order. The same method is used to pre set the output values (W (out) This allows to concentrate mainly on the interpolation properties of a well organized map. Figure 2 shows a comparison of the affine transformation with the projection method [G op93] In both cases an increase in the number of winners leads to a decrease in the root mean square error (RMS) The affine transformation behaves much better, especially for 4 to 10 winners. A topological choice of 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 RMS Error Number of winners Projection Method ....

J. G oppert and W. Rosenstiel. Topology-preserving interpolation in self-organizing maps. In Proceedings of NeuroNimes 93, pages 425--434, Nanterre, France, 10 1993. EC2.

....of optimal data topology, but without topology preservation. The use of interpolation techniques in the output layer may increase the performance of the system once more. Interpolation produces real output values and allow to work with SOM s of reduced size. Two different interpolation methods [1] using geometrical and topological interpolation are used. 2 Used Architecture In this work we are using the normal counter propagation architecture with a self organizing map in the competition layer. A schematic representation is shown in figure 1. Input (out) out) out) 1 2 m y y y x x x ....

....to improve the output values [2] During the recognition phase an input vector activates the best matching input weight vector W (in) i at neuron i, which applies the corresponding output vector W (out) i to the network output. 2. 2 Winner takes all WTA and two different interpolation methods [1] are used in this paper. Geometrical interpolation tries to transfer the geometry in the input space into the space of output values. In the case of topology preserving map (SOM) ambiguous configurations can be avoided using the topology, which leads to a topological interpolation [1] The ....

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J. Goppert and W. Rosenstiel. Topology-preserving interpolation in Self-Organizing maps. In Proceedings of NeuroNimes 93, pages 425--434, Nanterre, France, 10 1993, EC2.

....of topological neighbours is an advantage, because the structure of the interpolation can be predefined. On the other hand topological defects lead also to big errors in the interpolation. 2. 1 Interpolation parameters by projection The first interpolation method was presented in the basic paper [Gop93]. It consists of an orthogonal projection of the error vector (from the actual approximation to the exact input) onto the distance vector from the actual approximation to the next winner (figure 2) The following procedure is iteratively repeated for all winners (1 i k) X 0 = W (in) w0 ; ....

J. Goppert and W. Rosenstiel. Topology-preserving interpolation in self-organizing maps. In Proceedings of NeuroNimes 93, pages 425--434, Nanterre, France, 10 1993. EC2.

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