Although these two approaches are the most obvious, Allwein et al. [Allwein et a1., 2000]

We introduce the Concave-Convex procedure (CCCP) which constructs discretetime iterative dynamical systems which are guaranteed to monotonically decrease global optimization/energy functions. It can be applied to (almost) any optimization problem and many existing algorithms can be interpreted in terms of CCCP. In particular, we prove relationships to some applications of Legendre transform techniques. We then illustrate CCCP by applications to Potts models, linear assignment, EM algorithms, and Generalized Iterative Scaling (GIS). CCCP can be used both as a new way to understand existing optimization algorithms and as a procedure for generating new algorithms. 1 Introduction There is a lot of interest in designing discrete time dynamical systems for inference and learning (see, for example, [10], [3], [7], [13]).

Agakov System Engineering Research Group Chair of Manufacturing Technology Universitiit Erlangen-Niirnberg 91058 Erlangen, Germany F.Agakov@lft·uni-erlangen.de Stephen N. Felderhof Division of Informatics University of Edinburgh Edinburgh EH1 2QL, UK stephenf@dai.ed.ac.uk Abstract 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. Below weconsider PoE models in which each expert is a Gaussian. Although the product of Gaussians is also a Gaussian, if each Gaussian hasa 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.

There are two linear transformations to be considered, one operating inside thechannels (0) and one operating between the different channels (W). The two transformations are estimated in two adjacent leA steps. There are mainly two advantages, that can be taken from the first transformation: (i) By arranging independence among the columns of the transformed patches, the average transinformation betweendifferent channels is decreased.

The marriage of Renyi entropy with Parzen density estimation has been shown to be a viable tool in learning discriminative feature transforms. However, it suffers from computational complexity proportional to the square of the number of samples in the training data. This sets a practical limit to using large databases. We suggest immediate divorce of the two methods and remarriage of Renyi entropy with a semi-parametric density estimation method, such as a Gaussian Mixture Models (GMM). This allows allof the computation to take place in the low dimensional target space, and it reduces computational complexity proportional to square of the number of components in the mixtures. Furthermore, a convenient extensionto Hidden Markov Models as commonly used in speech recognition becomes possible.

We propose a new particle filter that incorporates a model of costs when generating particles. The approach is motivated by the observation that the costs of accidentally not tracking hypotheses might be significant in some areas of state space, and next to irrelevant in others. By incorporating acost model into particle filtering, states that are more critical to the system performance are more likely to be tracked. Automatic calculation of the cost model is implemented using an MDP value function calculation thatestimates the value of tracking a particular state. Experiments in two mobile robot domains illustrate the appropriateness of the approach.

To classify a large number of unlabeled examples we combine a limited numberof labeled examples with a Markov random walk representation overthe unlabeled examples.

This paper proposes an approach to classification of adjacent segments of a time series as being either of classes. We use a hierarchical model that consists of a feature extraction stage and a generative classifier which is built on top of these features. Such two stage approaches are often used in signal and image processing. The novel part of our work is that we link these stages probabilistically by using a latent feature space. To use one joint model is a Bayesian requirement, which has the advantage to fuse information according to its certainty.

We investigate a learning algorithm for the classification of nonnegative data by mixture models. Multiplicative update rules are derived that directly optimize the performance of these models as classifiers. The update rules have a simple closed form and an intuitive appeal. Our algorithm retains the main virtues of the Expectation-Maximization (EM) algorithm--its guarantee of monotonic improvement, andits absence of tuning parameters--with the added advantage of optimizing a discriminative objective function. The algorithm reduces as a special caseto the method of generalized iterative scaling for log-linear models. The learning rate of the algorithm is controlled by the sparseness of the training data. We use the method of nonnegative matrix factorization (NMF) to discover sparse distributed representations of the data. This form of feature selection greatly accelerates learning and makes the algorithm practical on large problems. Experiments showthat discriminatively trained mixture models lead to much better classification than comparably sized models trained by EM.

High dimensional data that lies on or near a low dimensional manifold can be described bya 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 coordination" ofthese 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 coordinate systemsare aligned in a consistent way. As a result, the internal coordinates changesmoothly and continuously as one traverses a connected path on the manifold--even when the path crosses the domains of many different local models. The regularizer takes the form of a Kullback-Leibler divergence and illustrates an unexpected application of variational methods: not to perform approximate inferencein intractable probabilistic models, but to learn more useful internal representations in tractable ones.