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Dimension reduction for high-dimensional compositional data plays an important role in many fields, where the principal component analysis of the basis covariance matrix is of scientific interest. In practice, however, the basis variables are latent and rarely observed, and standard techniques of principal component analysis are inadequate for compositional data because of the simplex constraint. To address the challenging problem, we relate the principal subspace of the centered log-ratio compositional covariance to that of the basis covariance, and prove that the latter is approximately identifiable with the diverging dimensionality under some subspace sparsity assumption. The interesting blessing-of-dimensionality phenomenon enables us to propose the principal subspace estimation methods by using the sample centered log-ratio covariance. We also derive nonasymptotic error bounds for the subspace estimators, which exhibits a tradeoff between identification and estimation. Moreover, we develop efficient proximal alternating direction method of multipliers algorithms to solve the nonconvex and nonsmooth optimization problems. Simulation results demonstrate that the proposed methods perform as well as the oracle methods with known basis. Their usefulness is illustrated through an analysis of word usage pattern for statisticians.

Principal component analysis (PCA) is a well-known linear dimension-reduction method that has been widely used in data analysis and modeling. It is an unsupervised learning technique that identifies a suitable linear subspace for the input variable that contains maximal variation and preserves as much information as possible. PCA has also been used in prediction models where the original, high-dimensional space of predictors is reduced to a smaller, more manageable, set before conducting regression analysis. However, this approach does not incorporate information in the response during the dimension-reduction stage and hence can have poor predictive performance. To address this concern, several supervised linear dimension-reduction techniques have been proposed in the literature. This paper reviews selected techniques, extends some of them, and compares their performance through simulations. Two of these techniques, partial least squares (PLS) and least-squares PCA (LSPCA), consistently outperform the others in this study.

In this paper, we develop a general framework to design differentially private expectation-maximization (EM) algorithms in high-dimensional latent variable models, based on the noisy iterative hard-thresholding. We derive the statistical guarantees of the proposed framework and apply it to three specific models: Gaussian mixture, mixture of regression, and regression with missing covariates. In each model, we establish the near-optimal rate of convergence with differential privacy constraints, and show the proposed algorithm is minimax rate optimal up to logarithm factors. The technical tools developed for the high-dimensional setting are then extended to the classic low-dimensional latent variable models, and we propose a near rate-optimal EM algorithm with differential privacy guarantees in this setting. Simulation studies and real data analysis are conducted to support our results.

Insights into complex, high-dimensional data can be obtained by discovering features of the data that match or do not match a model of interest. To formalize this task, we introduce the "data selection" problem: finding a lower-dimensional statistic - such as a subset of variables - that is well fit by a given parametric model of interest. A fully Bayesian approach to data selection would be to parametrically model the value of the statistic, nonparametrically model the remaining "background" components of the data, and perform standard Bayesian model selection for the choice of statistic. However, fitting a nonparametric model to high-dimensional data tends to be highly inefficient, statistically and computationally. We propose a novel score for performing both data selection and model selection, the "Stein volume criterion", that takes the form of a generalized marginal likelihood with a kernelized Stein discrepancy in place of the Kullback-Leibler divergence. The Stein volume criterion does not require one to fit or even specify a nonparametric background model, making it straightforward to compute - in many cases it is as simple as fitting the parametric model of interest with an alternative objective function. We prove that the Stein volume criterion is consistent for both data selection and model selection, and we establish consistency and asymptotic normality (Bernstein-von Mises) of the corresponding generalized posterior on parameters. We validate our method in simulation and apply it to the analysis of single-cell RNA sequencing datasets using probabilistic principal components analysis and a spin glass model of gene regulation.

In more and more applications, a quantity of interest may depend on several covariates, with at least one of them infinite-dimensional (e.g. a curve). To select the relevant covariates in this context, we propose an adaptation of the Lasso method. Two estimation methods are defined. The first one consists in the minimisation of a criterion inspired by classical Lasso inference under group sparsity (Yuan and Lin, 2006; Lounici et al., 2011) on the whole multivariate functional space H. The second one minimises the same criterion but on a finite-dimensional subspace of H which dimension is chosen by a penalized leasts-squares method base on the work of Barron et al. (1999). Sparsity-oracle inequalities are proven in case of fixed or random design in our infinite-dimensional context. To calculate the solutions of both criteria, we propose a coordinate-wise descent algorithm, inspired by the glmnet algorithm (Friedman et al., 2007). A numerical study on simulated and experimental datasets illustrates the behavior of the estimators.

We introduce and illustrate through numerical examples the R package \texttt{SIHR} which handles the statistical inference for (1) linear and quadratic functionals in the high-dimensional linear regression and (2) linear functional in the high-dimensional logistic regression. The focus of the proposed algorithms is on the point estimation, confidence interval construction and hypothesis testing. The inference methods are extended to multiple regression models. We include real data applications to demonstrate the package's performance and practicality.

A rich class of network models associate each node with a low-dimensional latent coordinate that controls the propensity for connections to form. Models of this type are well established in the literature, where it is typical to assume that the underlying geometry is Euclidean. Recent work has explored the consequences of this choice and has motivated the study of models which rely on non-Euclidean latent geometries, with a primary focus on spherical and hyperbolic geometry. In this paper\footnote{This is the first version of this work. Any potential mistake belongs to the first author.}, we examine to what extent latent features can be inferred from the observable links in the network, considering network models which rely on spherical, hyperbolic and lattice geometries. For each geometry, we describe a latent network model, detail constraints on the latent coordinates which remove the well-known identifiability issues, and present schemes for Bayesian estimation. Thus, we develop a computational procedures to perform inference for network models in which the properties of the underlying geometry play a vital role. Furthermore, we access the validity of those models with real data applications.

Self-training algorithms, which train a model to fit pseudolabels predicted by another previously-learned model, have been very successful for learning with unlabeled data using neural networks. However, the current theoretical understanding of self-training only applies to linear models. This work provides a unified theoretical analysis of self-training with deep networks for semi-supervised learning, unsupervised domain adaptation, and unsupervised learning. At the core of our analysis is a simple but realistic ``expansion'' assumption, which states that a low-probability subset of the data must expand to a neighborhood with large probability relative to the subset. We also assume that neighborhoods of examples in different classes have minimal overlap. We prove that under these assumptions, the minimizers of population objectives based on self-training and input-consistency regularization will achieve high accuracy with respect to ground-truth labels. By using off-the-shelf generalization bounds, we immediately convert this result to sample complexity guarantees for neural nets that are polynomial in the margin and Lipschitzness. Our results help explain the empirical successes of recently proposed self-training algorithms which use input consistency regularization.

We study the impact of neural networks in text classification. Our focus is on training deep neural networks with proper weight initialization and greedy layer-wise pretraining. Results are compared with 1-layer neural networks and Support Vector Machines. We work with a dataset of labeled messages from the Twitter microblogging service and aim to predict weather conditions. A feature extraction procedure specific for the task is proposed, which applies dimensionality reduction using Latent Semantic Analysis. Our results show that neural networks outperform Support Vector Machines with Gaussian kernels, noticing performance gains from introducing additional hidden layers with nonlinearities. The impact of using Nesterov's Accelerated Gradient in backpropagation is also studied. We conclude that deep neural networks are a reasonable approach for text classification and propose further ideas to improve performance.

Deep learning has emerged as a powerful machine learning technique that learns multiple layers of representations or features of the data and produces state-of-the-art prediction results. Along with the success of deep learning in many other application domains, deep learning is also popularly used in sentiment analysis in recent years. This paper first gives an overview of deep learning and then provides a comprehensive survey of its current applications in sentiment analysis.