Investments can be spread over many possible assets to avoid risk (at the cost of obtaining only an average performance) or it can be focused on clusters of only a few promising assets (at the cost of increased risk). A trade-off between these two objectives can be reached by using the self-organizing map (SOM), a neural network paradigm which achieves a clustering of data points while simultaneously preserving their inherent neighborhood relations (topography). This amounts to a combination of clustering with local smoothing. In a case study involving investments in emerging stock markets the author illustrates the application of SOMs in investment decisions, with an improvement of about 30% in returns over other, more simple investment strategies.

We analyze the pattern formation behavior of a high&hyphendimensional self&hyphenorganizing map (SOM) model for the competitive projection of ON&hyphencenter&hyphentype and OFF&hyphencenter&hyphentype inputs to a common map layer. We mathematically show, and numerically confirm, that even isotropic stimuli can drive the development of oriented receptive fields and an orientation map in this model. This result provides an important missing link in the spectrum of pattern formation behaviors observed in SOM models. Extending the model by including further layers for binocular inputs, we also investigate the combined development of orientation and ocular dominance maps. A parameter region for combined patterns exists; corresponding maps show a preference for perpendicular intersection angles between iso&hyphenorientation lines and ocularity domain boundaries, consistent with experimental observations.

Neural maps project data from an input space onto a neuron position in a (often lower dimensional) output space grid in a neighborhood preserving way, with neighboring neurons in the output space responding to neighboring data points in the input space. A map-learning algorithm can achieve an optimal neighborhood preservation only, if the output space topology roughly matches the effective structure of the data in the input space. We here present a growth algorithm, called the GSOM or growing self-organizing map, which enhances a widespread map self-organization process, Kohonen's self-organizing feature map (SOFM), by an adaptation of the output space grid during learning. The GSOM restricts the output space structure to the shape of a general hypercubical shape, with the overall dimensionality of the grid and its extensions along the different directions being subject of the adaptation. This constraint meets the demands of many larger information processing systems, of which the neural map can be a part. We apply our GSOM-algorithm to three examples, two of which involve real world data. Using recently developed methods for measuring the degree of neighborhood preservation in neural maps, we find the GSOM-algorithm to produce maps which preserve neighborhoods in a nearly optimal fashion.

The magnification exponents μ occurring in adaptive map formation algorithms like Kohonen's self-organizing feature map deviate for the information theoretically optimal value μ = 1 as well as from the values that optimize, e.g., the mean square distortion error (μ = 1/3 for one-dimensional maps). At the same time, models for categorical perception such as the "perceptual magnet" effect, which are based on topographic maps, require negative magnification exponents μ < 0. We present an extension of the self-organizing feature map algorithm, which utilizes adaptive local learning step sizes to actually control the magnification properties of the map. By change of a single parameter, maps with optimal information transfer, with various minimal reconstruction errors, or with an inverted magnification can be generated. Analytic results on this new algorithm are complemented by numerical simulations.

It has been hypothesized that the different appearance of ocular dominance bands in the cat and the monkey is a consequence of the different mapping geometries in these species (LeVay et al. 1985; Anderson et al. 1988). Here I investigate the impact of areal geometries on the preferred direction of ocular dominance bands in two adaptive map formation models, the self-organizing feature map and the elastic net algorithm. In the case of the self-organizing feature map, the occurrence of instabilities that correspond to ocular dominance bands can be analytically investigated. The instabilities automatically yield stripes of correct orientation. These analytic results are complemented by simulations. In the case of the elastic net algorithm, simulations reveal two different parameter regimes of the algorithm, only one of which leads to stripes of correct orientation. The results suggest that neighborhood preservation in visual maps is enforced in the backward direction, such that neighboring cells in the cortex have neighboring receptive fields, and not vice versa.

It is shown that a topographic product P, first introduced in nonlinear dynamics, is an appropriate measure of the preservation or violation of neighborhood relations. It is sensitive to large-scale violations of the neighborhood ordering, but does not account for neighborhood ordering distortions caused by varying areal magnification factors. A vanishing value of the topographic product indicates a perfect neighborhood preservation; negative (positive) values indicate a too small (too large) output space dimensionality. In a simple example of maps from a 2D input space onto 1D, 2D, and 3D output spaces, it is demonstrated how the topographic product picks the correct output space dimensionality. In a second example, 19D speech data are mapped onto various output spaces and it is found that a 3D output space (instead of 2D) seems to be optimally suited to the data. This is an agreement with a recent speech recognition experiment on the same data set.

Neural networks for such perceptual tasks as speech recognition must provide even more invariances than nets dealing with static problems, e.g., invariance under presentation speed fluctuations. The authors presently show that multilayer perceptrons with feedback over several layers (FMLPs) can meet these requirements. FMLPs can be trained simply with the open-loop learning rule. An analytical criterion for the stability of the resulting feedback states is given. By optimizing the output pattern representation, the stability of the feedback states can be improved. In the same way, the basins of attraction of the stable states can be enlarged. The performance with respect to presentation speed fluctuations is demonstrated in an example using three coupled FMLPs. In a continuous input sequence of letters, online detection of words is achieved. even when the presentation speed fluctuates in a wide range

We present a model for motion and direction detection of moving pulses whose performance is independent of pulse velocity, size and shape. The input signal activates one row of instantaneous nodes and one row of time integrating input nodes acting as short-term memories. Motion detection is achieved locally by subnetworks which are trained with a synthetic training set using the backpropagation algorithm. The global network is constructed from these subnetworks, one for each position. We test its performance with different pulse shapes and sizes and find the response to be invariant in a window of pulse velocities an order of magnitude wide. The window can be shifted by adjusting the memory time of the input nodes.