March 2010

The problem of automatic feature selection/weighting in kernel methods is examined. We work on a formulation that optimizes both the weights of features and the parameters of the kernel model simultaneously, using $L_1$ regularization for feature selection. Under quite general choices of kernels, we prove that there exists a unique regularization path for this problem, that runs from 0 to a stationary point of the non-regularized problem. We propose an ODE-based homotopy method to follow this trajectory. By […]

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Bayesian principal component analysis (BPCA), a probabilistic reformulation of PCA with Bayesian model selection, is a systematic approach to determining the number of essential principal components (PCs) for data representation. However, it assumes that data are Gaussian distributed and thus it cannot handle all types of practical observations, e.g. integers and binary values. In this paper, we propose simple exponential family PCA (SePCA), a generalised family of probabilistic principal component analysers. SePCA employs exponential family distributions to handle […]

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We introduce a new method – the group Dantzig selector – for high dimensional sparse regression with group structure, which has a convincing theory about why utilizing the group structure can be beneficial. Under a group restricted isometry condition, we obtain a significantly improved nonasymptotic $ell_2$-norm bound over the basis pursuit or the Dantzig selector which ignores the group structure. To gain more insight, we also introduce a surprisingly simple and intuitive “sparsity oracle condition” to obtain a […]

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This paper addresses multidimensional tuning parameter selection in the context of “train-validate-test” and $K$-fold cross validation. A coarse grid search over tuning parameter space is used to initialize a descent method which then jointly optimizes over variables and tuning parameters. We study four regularized regression methods and develop the update equations for the corresponding descent algorithms. Experiments on both simulated and real-world datasets show that the method results in significant tuning parameter refinement.

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A new approach to regression regularization called the Pairwise Elastic Net is proposed. Like the Elastic Net, it simultaneously performs automatic variable selection and continuous shrinkage. In addition, the Pairwise Elastic Net encourages the grouping of strongly correlated predictors based on a pairwise similarity measure. We give examples of how the Pairwise Elastic Net can be used to achieve the objectives of Ridge regression, the Lasso, the Elastic Net, and Group Lasso. Finally, we present a coordinate descent […]

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We study contextual multi-armed bandit problems where the context comes from a metric space and the payoff satisfies a Lipschitz condition with respect to the metric. Abstractly, a contextual multi-armed bandit problem models a situation where, in a sequence of independent trials, an online algorithm chooses, based on a given context (side information), an action from a set of possible actions so as to maximize the total payoff of the chosen actions. The payoff depends on both the […]

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Some online algorithms for linear classification model the uncertainty in their weights over the course of learning. Modeling the full covariance structure of the weights can provide a significant advantage for classification. However, for high-dimensional, large-scale data, even though there may be many second-order feature interactions, it is computationally infeasible to maintain this covariance structure. To extend second-order methods to high-dimensional data, we develop low-rank approximations of the covariance structure. We evaluate our approach on both synthetic and […]

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We develop a Bayesian framework for supervised dimension reduction using a flexible nonparametric Bayesian mixture modeling approach. Our method retrieves the dimension reduction or d.r. subspace by utilizing a dependent Dirichlet process that allows for natural clustering for the data in terms of both the response and predictor variables. Formal probabilistic models with likelihoods and priors are given and efficient posterior sampling of the d.r. subspace can be obtained by a Gibbs sampler. As the posterior draws are […]

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Recent research has seen the proposal of several new inductive principles designed specifically to avoid the problems associated with maximum likelihood learning in models with intractable partition functions. In this paper, we study learning methods for binary restricted Boltzmann machines (RBMs) based on ratio matching and generalized score matching. We compare these new RBM learning methods to a range of existing learning methods including stochastic maximum likelihood, contrastive divergence, and pseudo-likelihood. We perform an extensive empirical evaluation across […]

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Markov Random Fields (MRFs) are an important class of probabilistic models which are used for density estimation, classification, denoising, and for constructing Deep Belief Networks. Every application of an MRF requires addressing its inference problem, which can be done using deterministic inference methods or using stochastic Markov Chain Monte Carlo methods. In this paper we introduce a new Markov Chain transition operator that updates all the variables of a pairwise MRF in parallel by using auxiliary Gaussian variables. […]

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