High dimensional data that lies on or near a low dimensional manifold can be de- scribed by a collection of local linear models. Such a description, however, does not provide a global parameterization of the manifold--arguably an important goal of unsupervised learning. In this paper, we show how to learn a collection of local linear models that solves this more difficult problem. Our local linear models are represented by a mixture of factor analyzers, and the "global coordi- nation" of these models is achieved by adding a regularizing term to the standard maximum likelihood objective function. The regularizer breaks a degeneracy in the mixture model's parameter space, favoring models whose internal coor- dinate systems are aligned in a consistent way.

Recently Hinton (1999) has introduced the Products of Experts (PoE) model in which several individual probabilistic models for data are combined to provide an overall model of the data. Be(cid:173) low we consider PoE models in which each expert is a Gaussian. Although the product of Gaussians is also a Gaussian, if each Gaus(cid:173) sian has a simple structure the product can have a richer structure. We examine (1) Products of Gaussian pancakes which give rise to probabilistic Minor Components Analysis, (2) products of I-factor PPCA models and (3) a products of experts construction for an AR(l) process. Recently Hinton (1999) has introduced the Products of Experts (PoE) model in which several individual probabilistic models for data are combined to provide an overall model of the data.

We derive an equivalence between AdaBoost and the dual of a convex optimization problem, showing that the only difference between mini- mizing the exponential loss used by AdaBoost and maximum likelihood for exponential models is that the latter requires the model to be normal- ized to form a conditional probability distribution over labels. In addi- tion to establishing a simple and easily understood connection between the two methods, this framework enables us to derive new regularization procedures for boosting that directly correspond to penalized maximum likelihood. Experiments on UCI datasets support our theoretical analy- sis and give additional insight into the relationship between boosting and logistic regression.

We present a probabilistic generative model for timing deviations in expressive music. The structure of the proposed model is equivalent to a switching state space model. We formu(cid:173) late two well known music recognition problems, namely tempo tracking and automatic transcription (rhythm quantization) as fil(cid:173) tering and maximum a posteriori (MAP) state estimation tasks. The inferences are carried out using sequential Monte Carlo in(cid:173) tegration (particle filtering) techniques. For this purpose, we have derived a novel Viterbi algorithm for Rao-Blackwellized particle fil(cid:173) ters, where a subset of the hidden variables is integrated out.

We present three ways of combining linear programming with the kernel trick to find value function approximations for reinforcement learning. One formulation is based on SVM regression; the second is based on the Bellman equation; and the third seeks only to ensure that good moves have an advantage over bad moves. All formu(cid:173) lations attempt to minimize the number of support vectors while fitting the data. Experiments in a difficult, synthetic maze problem show that all three formulations give excellent performance, but the advantage formulation is much easier to train. Unlike policy gradi(cid:173) ent methods, the kernel methods described here can easily'adjust the complexity of the function approximator to fit the complexity of the value function.

We are interested in the mechanisms by which individuals monitor and adjust their performance of simple cognitive tasks. We model a speeded discrimination task in which individuals are asked to classify a sequence of stimuli (Jones & Braver, 2001). Response conflict arises when one stimulus class is infrequent relative to another, resulting in more errors and slower reaction times for the infrequent class. How do control pro- cesses modulate behavior based on the relative class frequencies? We explain performance from a rational perspective that casts the goal of individuals as minimizing a cost that depends both on error rate and re- action time.

It appears that in constructing the motor commands to guide the arm toward a target, the brain relies on an internal model (IM) of the dynamics of the task that it learns through practice [1]. The IM is presumably a system that transforms a desired limb trajectory in sensory coordinates to motor commands. The motor commands in turn create the complex activation of muscles necessary to cause action. A major issue in motor control is to infer characteristics of the IM from the actions of subjects. Recently, we took a first step toward mathematically characterizing the IM's rep- resentation in the brain [2]. We analyzed the sequence of errors made by subjects on successive movements as they reached to targets while holding a robotic arm. The robot produced a force field and subjects learned to compensate for the field (presumably by constructing an IM) and eventually produced straight movements within the field. Our analysis sought to draw conclusions about the structure of the IM from the sequence of errors generated by the subjects.

We describe a computer system that provides a real-time musi(cid:173) cal accompaniment for a live soloist in a piece of non-improvised music for soloist and accompaniment. A Bayesian network is devel(cid:173) oped that represents the joint distribution on the times at which the solo and accompaniment notes are played, relating the two parts through a layer of hidden variables. The network is first con(cid:173) structed using the rhythmic information contained in the musical score. The network is then trained to capture the musical interpre(cid:173) tations of the soloist and accompanist in an off-line rehearsal phase. During live accompaniment the learned distribution of the network is combined with a real-time analysis of the soloist's acoustic sig(cid:173) nal, performed with a hidden Markov model, to generate a musi(cid:173) cally principled accompaniment that respects all available sources of knowledge.

Statistical learning and probabilistic inference techniques are used to in- fer the hand position of a subject from multi-electrode recordings of neu- ral activity in motor cortex. First, an array of electrodes provides train- ing data of neural firing conditioned on hand kinematics. We learn a non- parametric representation of this firing activity using a Bayesian model and rigorously compare it with previous models using cross-validation. Second, we infer a posterior probability distribution over hand motion conditioned on a sequence of neural test data using Bayesian inference. The learned firing models of multiple cells are used to define a non- Gaussian likelihood term which is combined with a prior probability for the kinematics.

The recent introduction of the'relevance vector machine' has effec(cid:173) tively demonstrated how sparsity may be obtained in generalised linear models within a Bayesian framework. Using a particular form of Gaussian parameter prior, 'learning' is the maximisation, with respect to hyperparameters, of the marginal likelihood of the data. This paper studies the properties of that objective func(cid:173) tion, and demonstrates that conditioned on an individual hyper(cid:173) parameter, the marginal likelihood has a unique maximum which is computable in closed form. It is further shown that if a derived'sparsity criterion' is satisfied, this maximum is exactly equivalent to'pruning' the corresponding parameter from the model.