We present an algorithm to overcome the local maxima problem in estimating the parameters of mixture models. It combines existing approaches from both EM and a robust fitting algorithm, RANSAC, to give a data-driven stochastic learning scheme. Minimal subsets of data points, sufficient to constrain the parameters of the model, are drawn from proposal densities to discover new regions of high likelihood. The proposal densities are learnt using EM and bias the sampling toward promising solutions. The algorithm is computationally efficient, as well as effective at escaping from local maxima. We compare it with alternative methods, including EM and RANSAC, on both challenging synthetic data and the computer vision problem of alpha-matting.

In many applications, good ranking is a highly desirable performance for a classifier. The criterion commonly used to measure the ranking quality of a classification algorithm is the area under the ROC curve (AUC). To report it properly, it is crucial to determine an interval of confidence for its value. This paper provides confidence intervals for the AUC based on a statistical and combinatorial analysis using only simple parameters such as the error rate and the number of positive and negative examples. The analysis is distribution-independent, it makes no assumption about the distribution of the scores of negative or positive examples. The results are of practical use and can be viewed as the equivalent for AUC of the standard confidence intervals given in the case of the error rate. They are compared with previous approaches in several standard classification tasks demonstrating the benefits of our analysis.

Given a directed graph in which some of the nodes are labeled, we investigate the question of how to exploit the link structure of the graph to infer the labels of the remaining unlabeled nodes. To that extent we propose a regularization framework for functions defined over nodes of a directed graph that forces the classification function to change slowly on densely linked subgraphs. A powerful, yet computationally simple classification algorithm is derived within the proposed framework. The experimental evaluation on real-world Web classification problems demonstrates encouraging results that validate our approach.

In this paper we propose an efficient algorithm for reducing a large mixture of Gaussians into a smaller mixture while still preserving the component structure of the original model; this is achieved by clustering (grouping) the components. The method minimizes a new, easily computed distance measure between two Gaussian mixtures that can be motivated from a suitable stochastic model and the iterations of the algorithm use only the model parameters, avoiding the need for explicit resampling of datapoints. We demonstrate the method by performing hierarchical clustering of scenery images and handwritten digits.

We study the problem of hierarchical classification when labels corresponding to partial and/or multiple paths in the underlying taxonomy are allowed. We introduce a new hierarchical loss function, the H-loss, implementing the simple intuition that additional mistakes in the subtree of a mistaken class should not be charged for. Based on a probabilistic data model introduced in earlier work, we derive the Bayes-optimal classifier for the H-loss. We then empirically compare two incremental approximations of the Bayes-optimal classifier with a flat SVM classifier and with classifiers obtained by using hierarchical versions of the Perceptron and SVM algorithms. The experiments show that our simplest incremental approximation of the Bayes-optimal classifier performs, after just one training epoch, nearly as well as the hierarchical SVM classifier (which performs best). For the same incremental algorithm we also derive an H-loss bound showing, when data are generated by our probabilistic data model, exponentially fast convergence to the H-loss of the hierarchical classifier based on the true model parameters.

Planning algorithms designed for deterministic worlds, such as A* search, usually run much faster than algorithms designed for worlds with uncertain action outcomes, such as value iteration. Real-world planning problems often exhibit uncertainty, which forces us to use the slower algorithms to solve them. Many real-world planning problems exhibit sparse uncertainty: there are long sequences of deterministic actions which accomplish tasks like moving sensor platforms into place, interspersed with a small number of sensing actions which have uncertain outcomes. In this paper we describe a new planning algorithm, called MCP (short for MDP Compression Planning), which combines A* search with value iteration for solving Stochastic Shortest Path problem in MDPs with sparse stochasticity. We present experiments which show that MCP can run substantially faster than competing planners in domains with sparse uncertainty; these experiments are based on a simulation of a ground robot cooperating with a helicopter to fill in a partial map and move to a goal location.

We present a semi-parametric latent variable model based technique for density modelling, dimensionality reduction and visualization. Unlike previous methods, we estimate the latent distribution non-parametrically which enables us to model data generated by an underlying low dimensional, multimodal distribution. In addition, we allow the components of latent variable models to be drawn from the exponential family which makes the method suitable for special data types, for example binary or count data. Simulations on real valued, binary and count data show favorable comparison to other related schemes both in terms of separating different populations and generalization to unseen samples.

In the analysis of natural images, Gaussian scale mixtures (GSM) have been used to account for the statistics of filter responses, and to inspire hierarchical cortical representational learning schemes. GSMs pose a critical assignment problem, working out which filter responses were generated by a common multiplicative factor. We present a new approach to solving this assignment problem through a probabilistic extension to the basic GSM, and show how to perform inference in the model using Gibbs sampling. We demonstrate the efficacy of the approach on both synthetic and image data. Understanding the statistical structure of natural images is an important goal for visual neuroscience. Neural representations in early cortical areas decompose images (and likely other sensory inputs) in a way that is sensitive to sophisticated aspects of their probabilistic structure. This structure also plays a key role in methods for image processing and coding. A striking aspect of natural images that has reflections in both top-down and bottom-up modeling is coordination across nearby locations, scales, and orientations. From a topdown perspective, this structure has been modeled using what is known as a Gaussian Scale Mixture model (GSM).

Dominant sets are a new graph-theoretic concept that has proven to be relevant in pairwise data clustering problems, such as image segmentation. They generalize the notion of a maximal clique to edgeweighted graphs and have intriguing, nontrivial connections to continuous quadratic optimization and spectral-based grouping. We address the problem of grouping out-of-sample examples after the clustering process has taken place. This may serve either to drastically reduce the computational burden associated to the processing of very large data sets, or to efficiently deal with dynamic situations whereby data sets need to be updated continually. We show that the very notion of a dominant set offers a simple and efficient way of doing this. Numerical experiments on various grouping problems show the effectiveness of the approach.

We consider the problem of deriving class-size independent generalization bounds for some regularized discriminative multi-category classification methods. In particular, we obtain an expected generalization bound for a standard formulation of multi-category support vector machines. Based on the theoretical result, we argue that the formulation 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.