We develop a novel approach for supervised learning based on adaptively partitioning the feature space into different regions and learning local region-specific classifiers. We formulate an empirical risk minimization problem that incorporates both partitioning and classification in to a single global objective. We show that space partitioning can be equivalently reformulated as a supervised learning problem and consequently any discriminative learning method can be utilized in conjunction with our approach. Nevertheless, we consider locally linear schemes by learning linear partitions and linear region classifiers. Locally linear schemes can not only approximate complex decision boundaries and ensure low training error but also provide tight control on over-fitting and generalization error. We train locally linear classifiers by using LDA, logistic regression and perceptrons, and so our scheme is scalable to large data sizes and high-dimensions. We present experimental results demonstrating improved performance over state of the art classification techniques on benchmark datasets. We also show improved robustness to label noise.

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 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.

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.

We consider the problem of cardinality penalized optimization of a convex function over the probability simplex with additional convex constraints. It's well-known that the classical L1 regularizer fails to promote sparsity on the probability simplex since L1 norm on the probability simplex is trivially constant. We propose a direct relaxation of the minimum cardinality problem and show that it can be efficiently solved using convex programming. As a first application we consider recovering a sparse probability measure given moment constraints, in which our formulation becomes linear programming, hence can be solved very efficiently. A sufficient condition for exact recovery of the minimum cardinality solution is derived for arbitrary affine constraints. We then develop a penalized version for the noisy setting which can be solved using second order cone programs. The proposed method outperforms known heuristics based on L1 norm. As a second application we consider convex clustering using a sparse Gaussian mixture and compare our results with the well known soft k-means algorithm.

Non-linear dynamical systems (DS) have been used extensively for building generative models of human behavior. Its applications range from modeling brain dynamics to encoding motor commands. Many schemes have been proposed for encoding robot motions using dynamical systems with a single attractor placed at a predefined target in state space. Although these enable the robots to react against sudden perturbations without any re-planning, the motions are always directed towards a single target. In this work, we focus on combining several such DS with distinct attractors, resulting in a multi-stable DS. We show its applicability in reach-to-grasp tasks where the attractors represent several grasping points on the target object. While exploiting multiple attractors provides more flexibility in recovering from unseen perturbations, it also increases the complexity of the underlying learning problem. Here we present the Augmented-SVM (A-SVM) model which inherits region partitioning ability of the well known SVM classifier and is augmented with novel constraints derived from the individual DS. The new constraints modify the original SVM dual whose optimal solution then results in a new class of support vectors (SV). These new SV ensure that the resulting multi-stable DS incurs minimum deviation from the original dynamics and is stable at each of the attractors within a finite region of attraction. We show, via implementations on a simulated 10 degrees of freedom mobile robotic platform, that the model is capable of real-time motion generation and is able to adapt on-the-fly to perturbations.

We describe a latent variable model for supervised dimensionality reduction and distance metric learning. The model discovers linear projections of high dimensional data that shrink the distance between similarly labeled inputs and expand the distance between differently labeled ones. The model’s continuous latent variables locate pairs of examples in a latent space of lower dimensionality. The model differs significantly from classical factor analysis in that the posterior distribution over these latent variables is not always multivariate Gaussian. Nevertheless we show that inference is completely tractable and derive an Expectation-Maximization (EM) algorithm for parameter estimation. We also compare the model to other approaches in distance metric learning. The model’s main advantage is its simplicity: at each iteration of the EM algorithm, the distance metric is re-estimated by solving an unconstrained least-squares problem. Experiments show that these simple updates are highly effective.