We address the problem of learning a symmetric positive definite matrix. The central issue is to design parameter updates that preserve positive definiteness. Our updates are motivated with the von Neumann diver- gence. Rather than treating the most general case, we focus on two key applications that exemplify our methods: On-line learning with a simple square loss and finding a symmetric positive definite matrix subject to symmetric linear constraints. The updates generalize the Exponentiated Gradient (EG) update and AdaBoost, respectively: the parameter is now a symmetric positive definite matrix of trace one instead of a probability vector (which in this context is a diagonal positive definite matrix with trace one).

We establish a mistake bound for an ensemble method for classification based on maximizing the entropy of voting weights subject to margin constraints. The bound is the same as a general bound proved for the Weighted Majority Algorithm, and similar to bounds for other variants of Winnow. We prove a more refined bound that leads to a nearly opti- mal algorithm for learning disjunctions, again, based on the maximum entropy principle. We describe a simplification of the on-line maximum entropy method in which, after each iteration, the margin constraints are replaced with a single linear inequality. The simplified algorithm, which takes a similar form to Winnow, achieves the same mistake bounds.

Spike sorting involves clustering spike trains recorded by a micro- electrode according to the source neuron. It is a complicated problem, which requires a lot of human labor, partly due to the non-stationary na- ture of the data. We propose an automated technique for the clustering of non-stationary Gaussian sources in a Bayesian framework. At a first search stage, data is divided into short time frames and candidate descrip- tions of the data as a mixture of Gaussians are computed for each frame. At a second stage transition probabilities between candidate mixtures are computed, and a globally optimal clustering is found as the MAP so- lution of the resulting probabilistic model. Transition probabilities are computed using local stationarity assumptions and are based on a Gaus- sian version of the Jensen-Shannon divergence.

Two players, Black and White, take turns to place stones on the intersections of an N N grid (usually N 19 but smaller boards are in use as well). All the stones of each player are identical. Players place their stones in order to create territory by occupying or surrounding areas of the board. The player with the most territory at the end of the game is the winner. A stone is captured if it has been completely surrounded (in the horizontal and vertical directions) by stones of the opponent's colour.

We consider the problem of deriving class-size independent generaliza- tion bounds for some regularized discriminative multi-category classi- fication methods. In particular, we obtain an expected generalization bound for a standard formulation of multi-category support vector ma- chines. Based on the theoretical result, we argue that the formula- tion over-penalizes misclassification error, which in theory may lead to poor generalization performance. A remedy, based on a generalization of multi-category logistic regression (conditional maximum entropy), is then proposed, and its theoretical properties are examined.

Visual action recognition is an important problem in computer vision. In this paper, we propose a new method to probabilistically model and recognize actions of articulated objects, such as hand or body gestures, in image sequences. Our method consists of three levels of representa- tion. At the low level, we first extract a feature vector invariant to scale and in-plane rotation by using the Fourier transform of a circular spatial histogram. Then, spectral partitioning [20] is utilized to obtain an initial clustering; this clustering is then refined using a temporal smoothness constraint.

Motivated by the particular problems involved in communicating with "locked-in" paralysed patients, we aim to develop a brain- computer interface that uses auditory stimuli. We describe a paradigm that allows a user to make a binary decision by focusing attention on one of two concurrent auditory stimulus sequences. Using Support Vector Machine classification and Recursive Chan- nel Elimination on the independent components of averaged event- related potentials, we show that an untrained user's EEG data can be classified with an encouragingly high level of accuracy. This suggests that it is possible for users to modulate EEG signals in a single trial by the conscious direction of attention, well enough to be useful in BCI.

Linear Discriminant Analysis (LDA) is a well-known scheme for feature extraction and dimension reduction. It has been used widely in many ap- plications involving high-dimensional data, such as face recognition and image retrieval. An intrinsic limitation of classical LDA is the so-called singularity problem, that is, it fails when all scatter matrices are singu- lar. A well-known approach to deal with the singularity problem is to apply an intermediate dimension reduction stage using Principal Com- ponent Analysis (PCA) before LDA. The algorithm, called PCA LDA, is used widely in face recognition.

A graph-based prior is proposed for parametric semi-supervised classi- fication. The prior utilizes both labelled and unlabelled data; it also in- tegrates features from multiple views of a given sample (e.g., multiple sensors), thus implementing a Bayesian form of co-training. An EM algorithm for training the classifier automatically adjusts the tradeoff be- tween the contributions of: (a) the labelled data; (b) the unlabelled data; and (c) the co-training information. Active label query selection is per- formed using a mutual information based criterion that explicitly uses the unlabelled data and the co-training information. Encouraging results are presented on public benchmarks and on measured data from single and multiple sensors.

In this paper we propose to combine two powerful ideas, boosting and manifold learning. On the one hand, we improve ADABOOST by incor- porating knowledge on the structure of the data into base classifier design and selection. On the other hand, we use ADABOOST's efficient learn- ing mechanism to significantly improve supervised and semi-supervised algorithms proposed in the context of manifold learning. Beside the spe- cific manifold-based penalization, the resulting algorithm also accommo- dates the boosting of a large family of regularized learning algorithms. ADABOOST [1] is one of the machine learning algorithms that have revolutionized pattern recognition technology in the last decade.