It has long been known that lateral inhibition in neural networks can lead to a winner-take-all competition, so that only a single neuron is active at a steady state. Here we show how to organize lateral inhibition so that groups of neurons compete to be active. Given a collection of potentially overlapping groups, the inhibitory connectivity is set by a formula that can be interpreted as arising from a simple learning rule. Our analysis demonstrates that such inhibition generally results in winner-take-all competition between the given groups, with the exception of some degenerate cases. In a broader context, the network serves as a particular illustration of the general distinction between permitted and forbidden sets, which was introduced recently.

We introduce stagewise processing in error-correcting codes and image restoration, by extracting information from the former stage and using it selectively to improve the performance of the latter one. Both mean-field analysis using the cavity method and simulations show that it has the advantage of being robust against uncertainties in hyperparameter estimation. 1 Introduction In error-correcting codes [1] and image restoration [2], the choice of the so-called hyperparameters is an important factor in determining their performances. Hyperparameters refer to the coefficients weighing the biases and variances of the tasks. In error correction, they determine the statistical significance given to the paritychecking terms and the received bits. Similarly in image restoration, they determine the statistical weights given to the prior knowledge and the received data.

Using statistical mechanics results, I calculate learning curves (average generalization error) for Gaussian processes (GPs) and Bayesian neural networks (NNs) used for regression. Applying the results to learning a teacher defined by a two-layer network, I can directly compare GP and Bayesian NN learning.

Algebraic geometry is essential to learning theory. In hierarchical learning machines such as layered neural networks and gaussian mixtures, the asymptotic normality does not hold, since Fisher information matrices are singular. In this paper, the rigorous asymptotic form of the stochastic complexity is clarified based on resolution of singularities and two different problems are studied.

We analyze Gallager codes by employing a simple mean-field approximation that distorts the model geometry and preserves important interactions between sites. The method naturally recovers the probability propagation decoding algorithm as an extremization of a proper free-energy. We find a thermodynamic phase transition that coincides with information theoretical upper-bounds and explain the practical code performance in terms of the free-energy landscape.

In this paper we give necessary and sufficient conditions under which kernels of dot product type k(x, y) k(x.

A method is described which, like the kernel trick in support vector machines (SVMs), lets us generalize distance-based algorithms to operate in feature spaces, usually nonlinearly related to the input space. This is done by identifying a class of kernels which can be represented as norm-based distances in Hilbert spaces. It turns out that common kernel algorithms, such as SVMs and kernel PCA, are actually really distance based algorithms and can be run with that class of kernels, too. As well as providing a useful new insight into how these algorithms work, the present work can form the basis for conceiving new algorithms. 1 Introduction One of the crucial ingredients of SVMs is the so-called kernel trick for the computation of dot products in high-dimensional feature spaces using simple functions defined on pairs of input patterns. This trick allows the formulation of nonlinear variants of any algorithm that can be cast in terms of dot products, SVMs being but the most prominent example [13, 8]. Although the mathematical result underlying the kernel trick is almost a century old [6], it was only much later [1, 3,13] that it was made fruitful for the machine learning community. Kernel methods have since led to interesting generalizations of learning algorithms and to successful real-world applications. The present paper attempts to extend the utility of the kernel trick by looking at the problem of which kernels can be used to compute distances in feature spaces. Again, the underlying mathematical results, mainly due to Schoenberg, have been known for a while [7]; some of them have already attracted interest in the kernel methods community in various contexts [11, 5, 15].

The Bayesian paradigm apparently only sometimes gives rise to Occam's Razor; at other times very large models perform well. We give simple examples of both kinds of behaviour. The two views are reconciled when measuring complexity of functions, rather than of the machinery used to implement them. We analyze the complexity of functions for some linear in the parameter models that are equivalent to Gaussian Processes, and always find Occam's Razor at work. 1 Introduction Occam's Razor is a well known principle of "parsimony of explanations" which is influential in scientific thinking in general and in problems of statistical inference in particular. In this paper we review its consequences for Bayesian statistical models, where its behaviour can be easily demonstrated and quantified.

Learning of a smooth but nonparametric probability density can be regularized using methods of Quantum Field Theory. We implement a field theoretic prior numerically, test its efficacy, and show that the free parameter of the theory (,smoothness scale') can be determined self consistently by the data; this forms an infinite dimensional generalization of the MDL principle. Finally, we study the implications of one's choice of the prior and the parameterization and conclude that the smoothness scale determination makes density estimation very weakly sensitive to the choice of the prior, and that even wrong choices can be advantageous for small data sets. One of the central problems in learning is to balance'goodness of fit' criteria against the complexity of models. An important development in the Bayesian approach was thus the realization that there does not need to be any extra penalty for model complexity: if we compute the total probability that data are generated by a model, there is a factor from the volume in parameter space-the'Occam factor' -that discriminates against models with more parameters [1, 2].

The problem of constructing weak classifiers for boosting algorithms is studied. We present an algorithm that produces a linear classifier that is guaranteed to achieve an error better than random guessing for any distribution on the data. While this weak learner is not useful for learning in general, we show that under reasonable conditions on the distribution it yields an effective weak learner for one-dimensional problems. Preliminary simulations suggest that similar behavior can be expected in higher dimensions, a result which is corroborated by some recent theoretical bounds. Additionally, we provide improved convergence rate bounds for the generalization error in situations where the empirical error can be made small, which is exactly the situation that occurs if weak learners with guaranteed performance that is better than random guessing can be established.