Over the last few years the Fourier Transform (FT) representation of boolean functions has been an instrumental tool in the computational learning theory community. It has been used mainly to demonstrate the learnability of various classes of functions with respect to the uniform distribution.

In this paper, recursive estimation algorithms for dynamic modular networks are developed. The models are based on Gaussian RBF networks and the gating network is considered in two stages: At first, it is simply a time-varying scalar and in the second, it is based on the state, as in the mixture of local experts scheme. The resulting algorithm uses Kalman filter estimation for the model estimation and the gating probability estimation. Both, 'hard' and'soft' competition based estimation schemes are developed where in the former, the most probable network is adapted and in the latter all networks are adapted by appropriate weighting of the data. 1 INTRODUCTION The problem of learning multiple modes in a complex nonlinear system is increasingly being studied by various researchers [2, 3, 4, 5, 6], The use of a mixture of local experts [5, 6], and a conditional mixture density network [3] have been developed to model various modes of a system. The development has mainly been on model estimation from a given set of block data, with the model likelihood dependent on the input to the networks.

In this paper we examine a perceptron learning task. The task is realizable since it is provided by another perceptron with identical architecture. Both perceptrons have nonlinear sigmoid output functions. The gain of the output function determines the level of nonlinearity of the learning task. It is observed that a high level of nonlinearity leads to overfitting. We give an explanation for this rather surprising observation and develop a method to avoid the overfitting. This method has two possible interpretations, one is learning with noise, the other cross-validated early stopping.

A statistical theory for overtraining is proposed. The analysis treats realizable stochastic neural networks, trained with Kullback Leibler loss in the asymptotic case. It is shown that the asymptotic gain in the generalization error is small if we perform early stopping, even if we have access to the optimal stopping time. Considering cross-validation stopping we answer the question: In what ratio the examples should be divided into training and testing sets in order to obtain the optimum performance. In the non-asymptotic region cross-validated early stopping always decreases the generalization error. Our large scale simulations done on a CM5 are in nice agreement with our analytical findings.

The center-surround opponent receptive field (CSRF) mechanism represents one such example. Here, analogous CSRFs are shown to be formed in an artificial neural network which learns to localize contours (edges) of the luminance difference. Furthermore, when the input pattern is corrupted by a background noise, the CSRFs of the hidden units becomes shallower and broader with decrease of the signal-to-noise ratio (SNR). The same kind of SNR-dependent plasticity is present in the CSRF of real visual neurons; in bipolar cells of the carp retina as is shown here experimentally, as well as in large monopolar cells of the fly compound eye as was described by others. Also, analogous SNRdependent plasticity is shown to be present in the biphasic flash responses (BPFR) of these artificial and biological visual systems. Thus, the spatial (CSRF) and temporal (BPFR) filtering properties with which a wide variety of creatures see the world appear to be optimized for detectability of changes in space and time. 1 INTRODUCTION A number of learning algorithms have been developed to make synthetic neural machines be trainable to function in certain optimal ways. If the brain and nervous systems that we see in nature are best answers of the evolutionary process, then one might be able to find some common'softwares' in real and artificial neural systems. This possibility is examined in this paper, with respect to a basic visual 160 S. Y ASUI, T. FURUKAWA, M. YAMADA, T. SAITO

The existence of significant delays in sensorimotor feedback pathways has led several researchers to suggest that the cerebellum might function as a forward model of the motor plant in order to predict the sensory consequences of motor commands before actual feedback is available; e.g., (Ito, 1984; Keeler, 1990; Miall et ai., 1993). While we agree that there are many potential roles for forward models in motor control systems, as discussed, e.g., in (Wolpert et al., 1995), we present a hypothesis about how the cerebellum could participate in regulating movement in the presence of significant feedback delays without resorting to a forward model. We show how a very simplified version of the adjustable pattern generator (APG) model being developed by Houk and colleagues (Berthier et al., 1993; Houk et al., 1995) can learn to control endpoint positioning of a nonlinear spring-mass system with significant delays in both afferent and efferent pathways. Although much simpler than a multilink dynamic arm, control of this spring-mass system involves some of the challenges critical in the control of a more realistic motor system and serves to illustrate the principles we propose. Preliminary results appear in (Buckingham et al., 1995).

Various parameterizations of rotations are related through a unifying mathematical treatment, and transformations between coordinate systems are computed using the Campbell-Baker Hausdorff formula. Next, we describe Listing's law by means of the Lie algebra so(3). This enables us to demonstrate a direct connection to Donders' law, by showing that eye orientations are restricted to the quotient space 80(3)/80(2). The latter is equivalent to the sphere S2, which is exactly the space of gaze directions. Our analysis provides a mathematical framework for studying the oculomotor system and could also be extended to investigate the geometry of mUlti-joint arm movements.

Most studies of attention have focused on the selection process of incoming sensory cues (Posner et al., 1980; Koch et al., 1985; Desimone et al., 1995). Emphasis was placed on the phenomena of causing different percepts for the same sensory stimuli. However, the selection of sensory input itself is not the final goal of attention. We consider attention as a means for goal-directed behavior and survival of the animal. In this view, dynamical properties of attention are crucial. While attention has to be maintained long enough to enable robust response to sensory input, it also has to be shifted quickly to a novel cue that is potentially important. Long-term maintenance and quick transition are critical requirements for attention dynamics.

We propose a new learning method, "Generalized Learning Vector Quantization (GLVQ)," in which reference vectors are updated based on the steepest descent method in order to minimize the cost function. The cost function is determined so that the obtained learning rule satisfies the convergence condition. We prove that Kohonen's rule as used in LVQ does not satisfy the convergence condition and thus degrades recognition ability. Experimental results for printed Chinese character recognition reveal that GLVQ is superior to LVQ in recognition ability.

We have developed a foveated gesture recognition system that runs in an unconstrained office environment with an active camera. Using visionroutines previously implemented for an interactive environment, wedetermine the spatial location of salient body parts of a user and guide an active camera to obtain images of gestures or expressions. A hidden-state reinforcement learning paradigm is used to implement visual attention. The attention module selects targets to foveate based on the goal of successful recognition, and uses a new multiple-model Q-Iearning formulation. Given a set of target and distractor gestures, our system can learn where to foveate to maximally discriminate a particular gesture. 1 INTRODUCTION Vision has numerous uses in the natural world.