We consider the high-dimensional heteroscedastic regression model, where the mean and the log variance are modeled as a linear combination of input variables. Existing literature on high-dimensional linear regres- sion models has largely ignored non-constant error variances, even though they commonly occur in a variety of applications ranging from biostatis- tics to finance. In this paper we study a class of non-convex penalized pseudolikelihood estimators for both the mean and variance parameters. We show that the Heteroscedastic Iterative Penalized Pseudolikelihood Optimizer (HIPPO) achieves the oracle property, that is, we prove that the rates of convergence are the same as if the true model was known. We demonstrate numerical properties of the procedure on a simulation study and real world data.

Computational indices related to n-gram production were developed in order to assess the potential for n-gram indices to predict human scores of essay quality. A regression analyses was conducted on a corpus of 313 argumentative essays. The analyses demonstrated that a variety of n-gram indices were highly correlated to essay quality, but were also highly correlated to the number of words in the text (although many of the n-gram indices were stronger predictors of writing quality than the number of words in a text). A second regression analysis was conducted on a corpus of 88 argumentative essays that were controlled for text length differences. This analysis demonstrated that n-gram indices were still strong predictors of essay quality when text length was not a factor.

Many studies have been conducted on how to detect emotion classes or magnitudes from multimedia information such as text, audio, and images. However, the methods that can use predicted emotion classes and magnitudes to render emotion expressions in Embodied Conversational Agents (ECA) are still unclear. This paper proposes a computer graphics methodology that uses predicted non-linear regression values to render facial expressions using mesh morphing techniques. Results of the rendering technique are presented and discussed.

The problem of joint feature selection across a group of related tasks has applications in many areas including biomedical informatics and computer vision. We consider the l2,1-norm regularized regression model for joint feature selection from multiple tasks, which can be derived in the probabilistic framework by assuming a suitable prior from the exponential family. One appealing feature of the l2,1-norm regularization is that it encourages multiple predictors to share similar sparsity patterns. However, the resulting optimization problem is challenging to solve due to the non-smoothness of the l2,1-norm regularization. In this paper, we propose to accelerate the computation by reformulating it as two equivalent smooth convex optimization problems which are then solved via the Nesterov's method-an optimal first-order black-box method for smooth convex optimization. A key building block in solving the reformulations is the Euclidean projection. We show that the Euclidean projection for the first reformulation can be analytically computed, while the Euclidean projection for the second one can be computed in linear time. Empirical evaluations on several data sets verify the efficiency of the proposed algorithms.

This paper presents a new algorithm for online linear regression whose efficiency guarantees satisfy the requirements of the KWIK (Knows What It Knows) framework. The algorithm improves on the complexity bounds of the current state-of-the-art procedure in this setting. We explore several applications of this algorithm for learning compact reinforcement-learning representations. We show that KWIK linear regression can be used to learn the reward function of a factored MDP and the probabilities of action outcomes in Stochastic STRIPS and Object Oriented MDPs, none of which have been proven to be efficiently learnable in the RL setting before. We also combine KWIK linear regression with other KWIK learners to learn larger portions of these models, including experiments on learning factored MDP transition and reward functions together.

Incorporating domain knowledge into the modeling process is an effective way to improve learning accuracy. However, as it is provided by humans, domain knowledge can only be specified with some degree of uncertainty. We propose to explicitly model such uncertainty through probabilistic constraints over the parameter space. In contrast to hard parameter constraints, our approach is effective also when the domain knowledge is inaccurate and generally results in superior modeling accuracy. We focus on generative and conditional modeling where the parameters are assigned a Dirichlet or Gaussian prior and demonstrate the framework with experiments on both synthetic and real-world data.

Learning the parameters of a (potentially partially observable) random field model is intractable in general. Instead of focussing on a single optimal parameter value we propose to treat parameters as dynamical quantities. We introduce an algorithm to generate complex dynamics for parameters and (both visible and hidden) state vectors. We show that under certain conditions averages computed over trajectories of the proposed dynamical system converge to averages computed over the data. Our "herding dynamics" does not require expensive operations such as exponentiation and is fully deterministic.

We consider the problem of learning a high-dimensional multi-task regression model, under sparsity constraints induced by presence of grouping structures on the input covariates and on the output predictors. This problem is primarily motivated by expression quantitative trait locus (eQTL) mapping, of which the goal is to discover genetic variations in the genome (inputs) that influence the expression levels of multiple co-expressed genes (outputs), either epistatically, or pleiotropically, or both. A structured input-output lasso (SIOL) model based on an intricate l1/l2-norm penalty over the regression coefficient matrix is employed to enable discovery of complex sparse input/output relationships; and a highly efficient new optimization algorithm called hierarchical group thresholding (HiGT) is developed to solve the resultant non-differentiable, non-separable, and ultra high-dimensional optimization problem. We show on both simulation and on a yeast eQTL dataset that our model leads to significantly better recovery of the structured sparse relationships between the inputs and the outputs, and our algorithm significantly outperforms other optimization techniques under the same model. Additionally, we propose a novel approach for efficiently and effectively detecting input interactions by exploiting the prior knowledge available from biological experiments.

In this paper we propose a novel framework for the construction of sparsity-inducing priors. In particular, we define such priors as a mixture of exponential power distributions with a generalized inverse Gaussian density (EP-GIG). EP-GIG is a variant of generalized hyperbolic distributions, and the special cases include Gaussian scale mixtures and Laplace scale mixtures. Furthermore, Laplace scale mixtures can subserve a Bayesian framework for sparse learning with nonconvex penalization. The densities of EP-GIG can be explicitly expressed. Moreover, the corresponding posterior distribution also follows a generalized inverse Gaussian distribution. These properties lead us to EM algorithms for Bayesian sparse learning. We show that these algorithms bear an interesting resemblance to iteratively re-weighted $\ell_2$ or $\ell_1$ methods. In addition, we present two extensions for grouped variable selection and logistic regression.

Department of Statistics, Rice University Abstract High-dimensional data common in genomics, proteomics, and chemometrics often contains complicated correlation structures. Recently, partial least squares (PLS) and Sparse PLS methods have gained attention in these areas as dimension reduction techniques in the context of supervised data analysis. We introduce a framework for Regularized PLS by solving a relaxation of the SIMPLS optimization problem with penalties on the PLS loadings vectors. Our approach enjoys many advantages including flexibility, general penalties, easy interpretation of results, and fast computation in high-dimensional settings. We also outline extensions of our methods leading to novel methods for Nonnegative PLS and Generalized PLS, an adaption of PLS for structured data. We demonstrate the utility of our methods through simulations and a case study on proton Nuclear Magnetic Resonance (NMR) spectroscopy data. To whom correspondence should be addressed; Department of Statistics, Rice University, MS 138, 6100 Main St., Houston, TX 77005 (email: gallen@rice.edu) 1 Introduction Technologies to measure high-throughput biomedical data in proteomics, chemometrics, and genomics have led to a proliferation of high-dimensional data that pose many statistical challenges. As genes, proteins, and metabolites, are biologically interconnected, the variables in these data sets are often highly correlated. In this context, several have recently advocated using partial least squares (PLS) for dimension reduction of supervised data, or data with a response or labels (Nguyen and Rocke, 2002b; Boulesteix and Strimmer, 2007; Rossouw et al., 2008; Chun and Keleş, 2010). First introduced by Wold (1966) as a regression method that uses least squares on a set of derived inputs accounting for multi-colinearities, others have since proposed alternative methods for PLS with multiple responses (de Jong, 1993) and for classification (Marx, 1996; Barker and Rayens, 2003).