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Capturing patterns of variation present in a dataset is important in exploratory data analysis and unsupervised learning. Contrastive dimension reduction methods, such as contrastive principal component analysis (cPCA), find patterns unique to a target dataset of interest by contrasting with a carefully chosen background dataset representing unwanted or uninteresting variation. However, such methods typically require a tuning parameter that governs the level of contrast, and it is unclear how to choose this parameter objectively. Furthermore, it is frequently of interest to contrast against multiple backgrounds, which is difficult to accomplish with existing methods. We propose unique component analysis (UCA), a tuning-free method that identifies low-dimensional representations of a target dataset relative to one or more comparison datasets. It is computationally efficient even with large numbers of features. We show in several experiments that UCA with a single background dataset achieves similar results compared to cPCA with various tuning parameters, and that UCA with multiple individual background datasets is superior to both cPCA with any single background data and cPCA with a pooled background dataset.

Nonparametric empirical Bayes methods provide a flexible and attractive approach to high-dimensional data analysis. One particularly elegant empirical Bayes methodology, involving the Kiefer-Wolfowitz nonparametric maximum likelihood estimator (NPMLE) for mixture models, has been known for decades. However, implementation and theoretical analysis of the Kiefer-Wolfowitz NPMLE are notoriously difficult. A fast algorithm was recently proposed that makes NPMLE-based procedures feasible for use in large-scale problems, but the algorithm calculates only an approximation to the NPMLE. In this paper we make two contributions. First, we provide upper bounds on the convergence rate of the approximate NPMLE's statistical error, which have the same order as the best known bounds for the true NPMLE. This suggests that the approximate NPMLE is just as effective as the true NPMLE for statistical applications. Second, we illustrate the promise of NPMLE procedures in a high-dimensional binary classification problem. We propose a new procedure and show that it vastly outperforms existing methods in experiments with simulated data. In real data analyses involving cancer survival and gene expression data, we show that it is very competitive with several recently proposed methods for regularized linear discriminant analysis, another popular approach to high-dimensional classification.

As the number of possible predictors generated by high-throughput experiments continues to increase, methods are needed to quickly screen out unimportant covariates. Model-based screening methods have been proposed and theoretically justified, but only for a few specific models. Model-free screening methods have also recently been studied, but can have lower power to detect important covariates. In this paper we propose EEScreen, a screening procedure that can be used with any model that can be fit using estimating equations, and provide unified results on its finite-sample screening performance. EEScreen thus generalizes many recently proposed model-based and model-free screening procedures. We also propose iEEScreen, an iterative version of EEScreen, and show that it is closely related to a recently studied boosting method for estimating equations. We show via simulations for two different estimating equations that EEScreen and iEEScreen are useful and flexible screening procedures, and demonstrate our methods on data from a multiple myeloma study.