An adaptive online algorithm is proposed to estimate hierarchical data structures for non-stationary data sources. The approach is based on the principle of minimum cross entropy to derive a decision tree for data clustering and it employs a metalearning idea (learning to learn) to adapt to changes in data characteristics. Its efficiency is demonstrated by grouping non-stationary artifical data and by hierarchical segmentation of LANDSAT images. 1 Introduction Unsupervised learning addresses the problem to detect structure inherent in unlabeled and unclassified data. N. The encoding usually is represented by an assignment matrix M (Mia), where Mia 1 if and only if Xi belongs to cluster L: 1 MiaV (Xi, Ya) measures the quality of a data partition, Le., optimal assignments and prototypes (M,y)OPt argminM,y1i (M,Y) minimize the inhomogeneity of clusters w.r.t. a given distance measure V. For reasons of simplicity we restrict the presentation to the ' sum-of-squared-error criterion V(x, y) To facilitate this minimization a deterministic annealing approach was proposed in [5] signments, which maps the discrete optimization problem, i.e. how to determine the data as via the Maximum Entropy Principle [2] to a continuous parameter es- Unsupervised Online Learning of Decision Trees for Data Analysis 515 timation problem.

We present a new approximate learning algorithm for Boltzmann Machines, using a systematic expansion of the Gibbs free energy to second order in the weights. The linear response correction to the correlations is given by the Hessian of the Gibbs free energy. The computational complexity of the algorithm is cubic in the number of neurons. We compare the performance of the exact BM learning algorithm with first order (Weiss) mean field theory and second order (TAP) mean field theory. The learning task consists of a fully connected Ising spin glass model on 10 neurons. We conclude that 1) the method works well for paramagnetic problems 2) the TAP correction gives a significant improvement over the Weiss mean field theory, both for paramagnetic and spin glass problems and 3) that the inclusion of diagonal weights improves the Weiss approximation for paramagnetic problems, but not for spin glass problems.

The generalization ability of a neural network can sometimes be improved dramatically by regularization. To analyze the improvement one needs more refined results than the asymptotic distribution of the weight vector. Here we study the simple case of one-dimensional linear regression under quadratic regularization, i.e., ridge regression. We study the random design, misspecified case, where we derive expansions for the optimal regularization parameter and the ensuing improvement. It is possible to construct examples where it is best to use no regularization.

Gaussian processes provide natural nonparametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both processes can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance.

Peter M Williams School of Cognitive and Computing Sciences University of Sussex Falmer, Brighton BN1 9QH, UK. email: peterw@cogs.susx.ac.uk Abstract This paper presents a new approach to the problem of modelling daily rainfall using neural networks. We first model the conditional distributions ofrainfall amounts, in such a way that the model itself determines the order of the process, and the time-dependent shape and scale of the conditional distributions. After integrating over particular weather patterns, weare able to extract seasonal variations and long-term trends. 1 Introduction Analysis of rainfall data is important for many agricultural, ecological and engineering activities. Design of irrigation and drainage systems, for instance, needs to take account not only of mean expected rainfall, but also of rainfall volatility. Estimates of crop yields also depend on the distribution of rainfall during the growing season, as well as on the overall amount.

The observed distribution of natural images is far from uniform. On the contrary, real images have complex and important structure thatcan be exploited for image processing, recognition and analysis. There have been many proposed approaches to the principled statisticalmodeling of images, but each has been limited in either the complexity of the models or the complexity of the images. Wepresent a nonparametric multi-scale statistical model for images that can be used for recognition, image de-noising, and in a "generative mode" to synthesize high quality textures.

Scale invariance is a fundamental property of ensembles of natural images[1]. Their non Gaussian properties [15, 16] are less well understood, but they indicate the existence of a rich statistical structure.In this work we present a detailed study of the marginal statistics of a variable related to the edges in the images. A numerical analysis shows that it exhibits extended self-similarity [3, 4, 5]. This is a scaling property stronger than self-similarity: all its moments can be expressed as a power of any given moment. More interesting, all the exponents can be predicted in terms of a multiplicative log-Poisson process. This is the very same model that was used very recently to predict the correct exponents of the structure functions of turbulent flows [6]. These results allow us to study the underlying multifractal singularities. In particular we find that the most singular structures are one-dimensional: the most singular manifold consists of sharp edges.

In contrast to the maximum likelihood approach which finds only a single estimate for the regression parameters, the Bayesian approach yields a distribution of weight parameters, p(wID), conditional on the training data D, and predictions are ex- ·Present address: SNN, University of Nijmegen, Geert Grooteplein 21, Nijmegen, The Netherlands.

Weconsider a deterministic system with continuous state and time with infinite horizon discounted cost functional.

Department of Physics, Cavendish Laboratory Cambridge University Abstract Until recently, artificial intelligence researchers have frowned upon the application of probability propagation in Bayesian belief networks thathave cycles. The probability propagation algorithm is only exact in networks that are cycle-free. However, it has recently been discovered that the two best error-correcting decoding algorithms areactually performing probability propagation in belief networks with cycles. 1 Communicating over a noisy channel Our increasingly wired world demands efficient methods for communicating bits of information over physical channels that introduce errors. Examples of real-world channels include twisted-pair telephone wires, shielded cable-TV wire, fiberoptic cable, deep-space radio, terrestrial radio, and indoor radio. Engineers attempt to correct the errors introduced by the noise in these channels through the use of channel coding which adds protection to the information source, so that some channel errors can be corrected.