Based on a simple convexity lemma, we develop bounds for different types of Bayesian prediction errors for regression with Gaussian processes. The basic bounds are formulated for a fixed training set. Simpler expressions are obtained for sampling from an input distribution which equals the weight function of the covariance kernel, yielding asymptotically tight results. The results are compared with numerical experiments.

We study the approximation of functions by two-layer feedforward neural networks, focusing on incremental algorithms which greedily add units, estimating single unit parameters at each stage. As opposed to standard algorithms for fixed architectures, the optimization at each stage is performed over a small number of parameters, mitigating many of the difficult numerical problems inherent in high-dimensional nonlinear optimization. We establish upper bounds on the error incurred by the algorithm, when approximating functions from the Sobolev class, thereby extending previous results which only provided rates of convergence for functions in certain convex hulls of functional spaces. By comparing our results to recently derived lower bounds, we show that the greedy algorithms are nearly optimal. Combined with estimation error results for greedy algorithms, a strong case can be made for this type of approach.

We consider recurrent analog neural nets where each gate is subject to Gaussian noise, or any other common noise distribution whose probability density function is nonzero on a large set.

However, because of the separation between excitation and inhibition, biological neural networks are asymmetrical. We study characteristic differences between asymmetrical networks and their symmetrical counterparts, showing that they have dramatically different dynamical behavior and also how the differences can be exploited for computational ends. We illustrate our results in the case of a network that is a selective amplifier.

AR-NNs are a natural generalization of the classic linear autoregressive AR(p) process (2) See, e.g., Brockwell & Davis (1987) for a comprehensive introduction into AR and ARMA (autoregressive moving average) models.

Following recent results [9, 8] showing the importance of the fatshattering dimension in explaining the beneficial effect of a large margin on generalization performance, the current paper investigates the implications of these results for the case of imbalanced datasets and develops two approaches to setting the threshold. The approaches are incorporated into ThetaBoost, a boosting algorithm for dealing with unequal loss functions. The performance of ThetaBoost and the two approaches are tested experimentally.

Recent works in parameter estimation and neural coding have demonstrated that optimal performance are related to the mutual information between parameters and data. We consider the mutual information in the case where the dependency in the parameter (a vector 8) of the conditional p.d.f. of each observation (a vector

Gaussian process (GP) prediction suffers from O(n3) scaling with the data set size n. By using a finite-dimensional basis to approximate the GP predictor, the computational complexity can be reduced. We derive optimal finite-dimensional predictors under a number of assumptions, and show the superiority of these predictors over the Projected Bayes Regression method (which is asymptotically optimal). We also show how to calculate the minimal model size for a given n. The calculations are backed up by numerical experiments.

We solve the dynamics of Hopfield-type neural networks which store sequences of patterns, close to saturation. The asymmetry of the interaction matrix in such models leads to violation of detailed balance, ruling out an equilibrium statistical mechanical analysis. Using generating functional methods we derive exact closed equations for dynamical order parameters, viz. the sequence overlap and correlation and response functions.

The kernel-parameter is one of the few tunable parameters in Support Vector machines, controlling the complexity of the resulting hypothesis. Its choice amounts to model selection and its value is usually found by means of a validation set. We present an algorithm which can automatically perform model selection with little additional computational cost and with no need of a validation set. In this procedure model selection and learning are not separate, but kernels are dynamically adjusted during the learning process to find the kernel parameter which provides the best possible upper bound on the generalisation error. Theoretical results motivating the approach and experimental results confirming its validity are presented.