In high-dimensional data, structured noise caused by observed and unobserved factors affecting multiple target variables simultaneously, imposes a serious challenge for modeling, by masking the often weak signal. Therefore, (1) explaining away the structured noise in multiple-output regression is of paramount importance. Additionally, (2) assumptions about the correlation structure of the regression weights are needed. We note that both can be formulated in a natural way in a latent variable model, in which both the interesting signal and the noise are mediated through the same latent factors. Under this assumption, the signal model then borrows strength from the noise model by encouraging similar effects on correlated targets. We introduce a hyperparameter for the \emph{latent signal-to-noise ratio} which turns out to be important for modelling weak signals, and an ordered infinite-dimensional shrinkage prior that resolves the rotational unidentifiability in reduced-rank regression models. Simulations and prediction experiments with metabolite, gene expression, FMRI measurement, and macroeconomic time series data show that our model equals or exceeds the state-of-the-art performance and, in particular, outperforms the standard approach of assuming independent noise and signal models.

In high-dimensional estimation, analysts are faced with more parameters $p$ than available observations $n$, and asymptotic analysis of performance allows the ratio $p/n\rightarrow \infty$. This situation makes regularization both necessary and desirable in order for estimators to possess theoretical guarantees. However, the amount of regularization, often determined by one or more tuning parameters, is integral to achieving good performance. In practice, choosing the tuning parameter is done through resampling methods (e.g. cross-validation), generalized information criteria, or reformulating the optimization problem (e.g. square-root lasso or scaled sparse regression). Each of these techniques comes with varying levels of theoretical guarantee for the low- or high-dimensional regimes. However, there are some notable deficiencies in the literature. The theory, and sometimes practice, of many methods relies on either the knowledge or estimation of the variance parameter, which is difficult to estimate in high dimensions. In this paper, we provide theoretical intuition suggesting that some previously proposed approaches based on information criteria work poorly in high dimensions. We introduce a suite of new risk estimators leveraging the burgeoning literature on high-dimensional variance estimation. Finally, we compare our proposal to many existing methods for choosing the tuning parameters for lasso regression by providing an extensive simulation to examine their finite sample performance. We find that our new estimators perform quite well, often better than the existing approaches across a wide range of simulation conditions and evaluation criteria.

This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the existence of the inverse. Second, such an inverse can be obtained in closed form. Invertibility is an important advantage over other learning methods, because it permits to understand the identified features in the input domain where the data has physical meaning. Moreover, it allows to evaluate the performance of dimensionality reduction in sensible (input-domain) units. Volume preservation also allows an easy computation of information theoretic quantities, such as the reduction in multi-information after the transform. Third, the analytical nature of PPA leads to a clear geometrical interpretation of the manifold: it allows the computation of Frenet-Serret frames (local features) and of generalized curvatures at any point of the space. And fourth, the analytical Jacobian allows the computation of the metric induced by the data, thus generalizing the Mahalanobis distance. These properties are demonstrated theoretically and illustrated experimentally. The performance of PPA is evaluated in dimensionality and redundancy reduction, in both synthetic and real datasets from the UCI repository.

This paper introduces a new unsupervised method for dimensionality reduction via regression (DRR). The algorithm belongs to the family of invertible transforms that generalize Principal Component Analysis (PCA) by using curvilinear instead of linear features. DRR identifies the nonlinear features through multivariate regression to ensure the reduction in redundancy between he PCA coefficients, the reduction of the variance of the scores, and the reduction in the reconstruction error. More importantly, unlike other nonlinear dimensionality reduction methods, the invertibility, volume-preservation, and straightforward out-of-sample extension, makes DRR interpretable and easy to apply. The properties of DRR enable learning a more broader class of data manifolds than the recently proposed Non-linear Principal Components Analysis (NLPCA) and Principal Polynomial Analysis (PPA). We illustrate the performance of the representation in reducing the dimensionality of remote sensing data. In particular, we tackle two common problems: processing very high dimensional spectral information such as in hyperspectral image sounding data, and dealing with spatial-spectral image patches of multispectral images. Both settings pose collinearity and ill-determination problems. Evaluation of the expressive power of the features is assessed in terms of truncation error, estimating atmospheric variables, and surface land cover classification error. Results show that DRR outperforms linear PCA and recently proposed invertible extensions based on neural networks (NLPCA) and univariate regressions (PPA).

A key question in modern statistics is how to make fast and reliable inferences for complex, high-dimensional data. While there has been much interest in sparse techniques, current methods do not generalize well to data with nonlinear structure. In this work, we present an orthogonal series estimator for predictors that are complex aggregate objects, such as natural images, galaxy spectra, trajectories, and movies. Our series approach ties together ideas from kernel machine learning, and Fourier methods. We expand the unknown regression on the data in terms of the eigenfunctions of a kernel-based operator, and we take advantage of orthogonality of the basis with respect to the underlying data distribution, P, to speed up computations and tuning of parameters. If the kernel is appropriately chosen, then the eigenfunctions adapt to the intrinsic geometry and dimension of the data. We provide theoretical guarantees for a radial kernel with varying bandwidth, and we relate smoothness of the regression function with respect to P to sparsity in the eigenbasis. Finally, using simulated and real-world data, we systematically compare the performance of the spectral series approach with classical kernel smoothing, k-nearest neighbors regression, kernel ridge regression, and state-of-the-art manifold and local regression methods.

Technology and collaboration enable dramatic increases in the size of psychological and psychiatric data collections, but finding structure in these large data sets with many collected variables is challenging. Decision tree ensembles like random forests (Strobl, Malley, and Tutz, 2009) are a useful tool for finding structure, but are difficult to interpret with multiple outcome variables which are often of interest in psychology. To find and interpret structure in data sets with multiple outcomes and many predictors (possibly exceeding the sample size), we introduce a multivariate extension to a decision tree ensemble method called Gradient Boosted Regression Trees (Friedman, 2001). Our method, multivariate tree boosting, can be used for identifying important predictors, detecting predictors with non-linear effects and interactions without specification of such effects, and for identifying predictors that cause two or more outcome variables to covary without parametric assumptions. We provide the R package 'mvtboost' to estimate, tune, and interpret the resulting model, which extends the implementation of univariate boosting in the R package 'gbm' (Ridgeway, 2013) to continuous, multivariate outcomes. To illustrate the approach, we analyze predictors of psychological well-being (Ryff and Keyes, 1995). Simulations verify that our approach identifies predictors with non-linear effects and achieves high prediction accuracy, exceeding or matching the performance of (penalized) multivariate multiple regression and multivariate decision trees over a wide range of conditions.

A generalized linear model (GLM) is a flexible extension of linear regression which allows the response or the output to be a nonlinear function of the input via an activation function. In other words, in a GLM, the linear regression of the input is passed through an activation function to generate the response. GLMs unify popular frameworks such as logistic regression and Poisson regression with linear regression. At the same time, they can be learnt with guarantees using simple iterative methods (Kakade et al., 2011). In many scenarios, however, GLMs may be too simplistic, and mixtures of GLMs can be much more effective since they combine the expressive power of latent variables with the predictive capabilities of the GLM. Mixtures of GLMs have widespread applicability including object recognition (Quattoni et al., 2004), human action recognition (Wang and Mori, 2009), syntactic parsing (Petrov and Klein, 2007), and machine translation (Liang et al., 2006). Traditionally, mixture models are learnt through heuristics such as expectation maximization (EM) (Jordan and Jacobs, 1994; Xu et al., 1995) or variational Bayes (Bishop and Svensen, 2003). However, these methods can converge to spurious local optima and have slow convergence rates for high dimensional models. In contrast, we employ a method-of-moments approach for guaranteed learning of mixtures of GLMs.

This paper proposes a distributionally robust approach to logistic regression. We use the Wasserstein distance to construct a ball in the space of probability distributions centered at the uniform distribution on the training samples. If the radius of this Wasserstein ball is chosen judiciously, we can guarantee that it contains the unknown data-generating distribution with high confidence. We then formulate a distributionally robust logistic regression model that minimizes a worst-case expected logloss function, where the worst case is taken over all distributions in the Wasserstein ball. We prove that this optimization problem admits a tractable reformulation and encapsulates the classical as well as the popular regularized logistic regression problems as special cases. We further propose a distributionally robust approach based on Wasserstein balls to compute upper and lower confidence bounds on the misclassification probability of the resulting classifier. These bounds are given by the optimal values of two highly tractable linear programs. We validate our theoretical out-of-sample guarantees through simulated and empirical experiments.

We propose a class of closed-form estimators for GLMs under high-dimensional sampling regimes. Our class of estimators is based on deriving closed-form variants of the vanilla unregularized MLE but which are (a) well-defined even under high-dimensional settings, and (b) available in closed-form. We then perform thresholding operations on this MLE variant to obtain our class of estimators. We derive a unified statistical analysis of our class of estimators, and show that it enjoys strong statistical guarantees in both parameter error as well as variable selection, that surprisingly match those of the more complex regularized GLM MLEs, even while our closed-form estimators are computationally much simpler. We derive instantiations of our class of closed-form estimators, as well as corollaries of our general theorem, for the special cases of logistic, exponential and Poisson regression models. We corroborate the surprising statistical and computational performance of our class of estimators via extensive simulations.

High dimensional regression benefits from sparsity promoting regularizations. Screening rules leverage the known sparsity of the solution by ignoring some variables in the optimization, hence speeding up solvers. When the procedure is proven not to discard features wrongly the rules are said to be safe. In this paper we derive new safe rules for generalized linear models regularized with L1 and L1/L2 norms. The rules are based on duality gap computations and spherical safe regions whose diameters converge to zero. This allows to discard safely more variables, in particular for low regularization parameters. The GAP Safe rule can cope with any iterative solver and we illustrate its performance on coordinate descent for multi-task Lasso, binary and multinomial logistic regression, demonstrating significant speed ups on all tested datasets with respect to previous safe rules.