Discriminative linear models are a popular tool in machine learning. These can be generally divided into two types: The first is linear classifiers, such as support vector machines, which are well studied and provide state-of-the-art results. One shortcoming of these models is that their output (known as the 'margin') is not calibrated, and cannot be translated naturally into a distribution over the labels. Thus, it is difficult to incorporate such models as components of larger systems, unlike probabilistic based approaches. The second type of approach constructs class conditional distributions using a nonlinearity (e.g. log-linear models), but is occasionally worse in terms of classification error. We propose a supervised learning method which combines the best of both approaches. Specifically, our method provides a distribution over the labels, which is a linear function of the model parameters. As a consequence, differences between probabilities are linear functions, a property which most probabilistic models (e.g. log-linear) do not have. Our model assumes that classes correspond to linear subspaces (rather than to half spaces). Using a relaxed projection operator, we construct a measure which evaluates the degree to which a given vector 'belongs' to a subspace, resulting in a distribution over labels. Interestingly, this view is closely related to similar concepts in quantum detection theory. The resulting models can be trained either to maximize the margin or to optimize average likelihood measures. The corresponding optimization problems are semidefinite programs which can be solved efficiently. We illustrate the performance of our algorithm on real world datasets, and show that it outperforms 2nd order kernel methods.

Latent feature models are attractive for image modeling, since images generally contain multiple objects. However, many latent feature models ignore that objects can appear at different locations or require pre-segmentation of images. While the transformed Indian buffet process (tIBP) provides a method for modeling transformation-invariant features in unsegmented binary images, its current form is inappropriate for real images because of its computational cost and modeling assumptions. We combine the tIBP with likelihoods appropriate for real images and develop an efficient inference, using the cross-correlation between images and features, that is theoretically and empirically faster than existing inference techniques. Our method discovers reasonable components and achieve effective image reconstruction in natural images.

Nonparametric Bayesian approaches to clustering, information retrieval, language modeling and object recognition have recently shown great promise as a new paradigm for unsupervised data analysis. Most contributions have focused on the Dirichlet process mixture models or extensions thereof for which efficient Gibbs samplers exist. In this paper we explore Gibbs samplers for infinite complexity mixture models in the stick breaking representation. The advantage of this representation is improved modeling flexibility. For instance, one can design the prior distribution over cluster sizes or couple multiple infinite mixture models (e.g. over time) at the level of their parameters (i.e. the dependent Dirichlet process model). However, Gibbs samplers for infinite mixture models (as recently introduced in the statistics literature) seem to mix poorly over cluster labels. Among others issues, this can have the adverse effect that labels for the same cluster in coupled mixture models are mixed up. We introduce additional moves in these samplers to improve mixing over cluster labels and to bring clusters into correspondence. An application to modeling of storm trajectories is used to illustrate these ideas.

We present a new approach to learning the structure and parameters of a Bayesian network based on regularized estimation in an exponential family representation. Here we show that, given a fixed variable order, the optimal structure and parameters can be learned efficiently, even without restricting the size of the parent sets. We then consider the problem of optimizing the variable order for a given set of features. This is still a computationally hard problem, but we present a convex relaxation that yields an optimal 'soft' ordering in polynomial time. One novel aspect of the approach is that we do not perform a discrete search over DAG structures, nor over variable orders, but instead solve a continuous relaxation that can then be rounded to obtain a valid network structure. We conduct an experimental comparison against standard structure search procedures over standard objectives, which cope with local minima, and evaluate the advantages of using convex relaxations that reduce the effects of local minima.

Multitask learning algorithms are typically designed assuming some fixed, a priori known latent structure shared by all the tasks. However, it is usually unclear what type of latent task structure is the most appropriate for a given multitask learning problem. Ideally, the "right" latent task structure should be learned in a data-driven manner. We present a flexible, nonparametric Bayesian model that posits a mixture of factor analyzers structure on the tasks. The nonparametric aspect makes the model expressive enough to subsume many existing models of latent task structures (e.g, mean-regularized tasks, clustered tasks, low-rank or linear/non-linear subspace assumption on tasks, etc.). Moreover, it can also learn more general task structures, addressing the shortcomings of such models. We present a variational inference algorithm for our model. Experimental results on synthetic and real-world datasets, on both regression and classification problems, demonstrate the effectiveness of the proposed method.

We show that the multi-class support vector machine (MSVM) proposed by Lee et. al. (2004), can be viewed as a MAP estimation procedure under an appropriate probabilistic interpretation of the classifier. We also show that this interpretation can be extended to a hierarchical Bayesian architecture and to a fully-Bayesian inference procedure for multi-class classification based on data augmentation. We present empirical results that show that the advantages of the Bayesian formalism are obtained without a loss in classification accuracy.

We present a hybrid algorithm for Bayesian topic models that combines the efficiency of sparse Gibbs sampling with the scalability of online stochastic inference. We used our algorithm to analyze a corpus of 1.2 million books (33 billion words) with thousands of topics. Our approach reduces the bias of variational inference and generalizes to many Bayesian hidden-variable models.

Collaborative data consist of ratings relating two distinct sets of objects: users and items. Much of the work with such data focuses on filtering: predicting unknown ratings for pairs of users and items. In this paper we focus on the problem of visualizing the information. Given all of the ratings, our task is to embed all of the users and items as points in the same Euclidean space. We would like to place users near items that they have rated (or would rate) high, and far away from those they would give a low rating. We pose this problem as a real-valued non-linear Bayesian network and employ Markov chain Monte Carlo and expectation maximization to find an embedding. We present a metric by which to judge the quality of a visualization and compare our results to local linear embedding and Eigentaste on three real-world datasets.

Continuous-time Bayesian networks (CTBNs) are graphical representations of multi-component continuous-time Markov processes as directed graphs. The edges in the network represent direct influences among components. The joint rate matrix of the multi-component process is specified by means of conditional rate matrices for each component separately. This paper addresses the situation where some of the components evolve on a time scale that is much shorter compared to the time scale of the other components. In this paper, we prove that in the limit where the separation of scales is infinite, the Markov process converges (in distribution, or weakly) to a reduced, or effective Markov process that only involves the slow components. We also demonstrate that for reasonable separation of scale (an order of magnitude) the reduced process is a good approximation of the marginal process over the slow components. We provide a simple procedure for building a reduced CTBN for this effective process, with conditional rate matrices that can be directly calculated from the original CTBN, and discuss the implications for approximate reasoning in large systems.

We propose a method to identify all the nodes that are relevant to compute all the conditional probability distributions for a given set of nodes. Our method is simple, effcient, consistent, and does not require learning a Bayesian network first. Therefore, our method can be applied to high-dimensional databases, e.g. gene expression databases.