A number of dependent nonparametric processes have been proposed to model non-stationary data with unknown latent dimensionality. However, the inference algorithms are often slow and unwieldy, and are in general highly specific to a given model formulation. In this paper, we describe a wide class of nonparametric processes, including several existing models, and present a slice sampler that allows efficient inference across this class of models.

We develop a method for discovering the parts of an articulated object from aligned meshes of the object in various three-dimensional poses. We adapt the distance dependentChinese restaurant process (ddCRP) to allow nonparametric discovery ofa potentially unbounded number of parts, while simultaneously guaranteeing a spatially connected segmentation. To allow analysis of datasets in which object instances have varying 3D shapes, we model part variability across poses via affine transformations. By placing a matrix normal-inverse-Wishart prior on these affine transformations, we develop a ddCRP Gibbs sampler which tractably marginalizes over transformation uncertainty. Analyzing a dataset of humans captured indozens of poses, we infer parts which provide quantitatively better deformation predictionsthan conventional clustering methods.

We develop a Bayesian nonparametric extension of the popular Plackett-Luce choice model that can handle an infinite number of choice items. Our framework is based on the theory of random atomic measures, with the prior specified by a gamma process. We derive a posterior characterization and a simple and effective Gibbs sampler for posterior simulation. We then develop a time-varying extension of our model, and apply our model to the New York Times lists of weekly bestselling books.

In Compressive Sensing Magnetic Resonance Imaging (CS-MRI), one can reconstruct a MR image with good quality from only a small number of measurements. This can significantly reduce MR scanning time. According to structured sparsity theory, the measurements can be further reduced to $\mathcal{O}(K+\log n)$ for tree-sparse data instead of $\mathcal{O}(K+K\log n)$ for standard $K$-sparse data with length $n$. However, few of existing algorithms has utilized this for CS-MRI, while most of them use Total Variation and wavelet sparse regularization. On the other side, some algorithms have been proposed for tree sparsity regularization, but few of them has validated the benefit of tree structure in CS-MRI. In this paper, we propose a fast convex optimization algorithm to improve CS-MRI. Wavelet sparsity, gradient sparsity and tree sparsity are all considered in our model for real MR images. The original complex problem is decomposed to three simpler subproblems then each of the subproblems can be efficiently solved with an iterative scheme. Numerous experiments have been conducted and show that the proposed algorithm outperforms the state-of-the-art CS-MRI algorithms, and gain better reconstructions results on real MR images than general tree based solvers or algorithms.

We address the problem of general supervised learning when data can only be accessed through an (indefinite) similarity function between data points. Existing work on learning with indefinite kernels has concentrated solely on binary/multiclass classification problems. We propose a model that is generic enough to handle any supervised learning task and also subsumes the model previously proposed for classification. We give a ''goodness'' criterion for similarity functions w.r.t. a given supervised learning task and then adapt a well-known landmarking technique to provide efficient algorithms for supervised learning using ''good'' similarity functions. We demonstrate the effectiveness of our model on three important supervised learning problems: a) real-valued regression, b) ordinal regression and c) ranking where we show that our method guarantees bounded generalization error. Furthermore, for the case of real-valued regression, we give a natural goodness definition that, when used in conjunction with a recent result in sparse vector recovery, guarantees a sparse predictor with bounded generalization error. Finally, we report results of our learning algorithms on regression and ordinal regression tasks using non-PSD similarity functions and demonstrate the effectiveness of our algorithms, especially that of the sparse landmark selection algorithm that achieves significantly higher accuracies than the baseline methods while offering reduced computational costs.

Mixture distributions are often used to model complex data. In this paper, we develop a new method that jointly estimates mixture models over multiple data sets by exploiting the statistical dependencies between them. Specifically, we introduce a set of latent Dirichlet processes as sources of component models (atoms), and for each data set, we construct a nonparametric mixture model by combining sub-sampled versions of the latent DPs. Each mixture model may acquire atoms from different latent DPs, while each atom may be shared by multiple mixtures. This multi-to-multi association distinguishes the proposed method from prior constructions that rely on tree or chain structures, allowing mixture models to be coupled more flexibly. In addition, we derive a sampling algorithm that jointly infers the model parameters and present experiments on both document analysis and image modeling.

This paper aims to take a step forwards making the term ``intrinsic motivation'' from reinforcement learning theoretically well founded, focusing on curiosity-driven learning. To that end, we consider the setting where, a fixed partition P of a continuous space X being given, and a process \nu defined on X being unknown, we are asked to sequentially decide which cell of the partition to select as well as where to sample \nu in that cell, in order to minimize a loss function that is inspired from previous work on curiosity-driven learning. The loss on each cell consists of one term measuring a simple worst case quadratic sampling error, and a penalty term proportional to the range of the variance in that cell. The corresponding problem formulation extends the setting known as active learning for multi-armed bandits to the case when each arm is a continuous region, and we show how an adaptation of recent algorithms for that problem and of hierarchical optimistic sampling algorithms for optimization can be used in order to solve this problem. The resulting procedure, called Hierarchical Optimistic Region SElection driven by Curiosity (HORSE.C) is provided together with a finite-time regret analysis.

We present a new formulation for attacking binary classification problems. Instead of relying on convex losses and regularisers such as in SVMs, logistic regression and boosting, or instead non-convex but continuous formulations such as those encountered in neural networks and deep belief networks, our framework entails a non-convex but \emph{discrete} formulation, where estimation amounts to finding a MAP configuration in a graphical model whose potential functions are low-dimensional discrete surrogates for the misclassification loss. We argue that such a discrete formulation can naturally account for a number of issues that are typically encountered in either the convex or the continuous non-convex paradigms, or both. By reducing the learning problem to a MAP inference problem, we can immediately translate the guarantees available for many inference settings to the learning problem itself. We empirically demonstrate in a number of experiments that this approach is promising in dealing with issues such as severe label noise, while still having global optimality guarantees. Due to the discrete nature of the formulation, it also allows for \emph{direct} regularisation through cardinality-based penalties, such as the $\ell_0$ pseudo-norm, thus providing the ability to perform feature selection and trade-off interpretability and predictability in a principled manner. We also outline a number of open problems arising from the formulation.

Stochastic differential equations (SDE) are a natural tool for modelling systems that are inherently noisy or contain uncertainties that can be modelled as stochastic processes. Crucial to the process of using SDE to build mathematical models is the ability to estimate parameters of those models from observed data. Over the past few decades, significant progress has been made on this problem, but we are still far from having a definitive solution. We describe a novel method of approximating a diffusion process that we show to be useful in Markov chain Monte-Carlo (MCMC) inference algorithms. We take the ‘white’ noise that drives a diffusion process and decompose it into two terms. The first is a ‘coloured noise’ term that can be deterministically controlled by a set of auxilliary variables. The second term is small and enables us to form a linear Gaussian ‘small noise’ approximation. The decomposition allows us to take a diffusion process of interest and cast it in a form that is amenable to sampling by MCMC methods. We explain why many state-of-the-art inference methods fail on highly nonlinear inference problems. We demonstrate experimentally that our method performs well in such situations. Our results show that this method is a promising new tool for use in inference and parameter estimation problems.

We study the scalability of consensus-based distributed optimization algorithms by considering two questions: How many processors should we use for a given problem, and how often should they communicate when communication is not free? Central to our analysis is a problem-specific value $r$ which quantifies the communication/computation tradeoff. We show that organizing the communication among nodes as a $k$-regular expander graph~\cite{kRegExpanders} yields speedups, while when all pairs of nodes communicate (as in a complete graph), there is an optimal number of processors that depends on $r$. Surprisingly, a speedup can be obtained, in terms of the time to reach a fixed level of accuracy, by communicating less and less frequently as the computation progresses. Experiments on a real cluster solving metric learning and non-smooth convex minimization tasks demonstrate strong agreement between theory and practice.