In the last decade, improvements in genomic, transcrip-tomic, and proteomic technologies have enabled personalized medicine (also called precision medicine) to become an essential part of contemporary medicine. Personalized medicine takes into account individual variability in genes, proteins, environment, and lifestyle to decide on optimal disease treatment and prevention [14]. The use of a patient's genetic and epigenetic information has already proven to be highly effective to tailor drug therapies or preventive care in a number of applications, such as breast [7], prostate [23], ovarian [17], and pancreatic cancers [24], cardiovascular disease [11], cystic fibrosis [36], and psychiatry [10]. The subfield of pharmacogenomics studies specifically how genes affect a person's response to particular drugs to develop more efficient and safer medications [37]. Genomic, epigenomic, and transcriptomic data used in precision medicine, such as gene expression, copy number variants, or methylation levels are typically high-dimensional with a number of variables that rivals or exceeds the number of observations.

The abundance of high-dimensional data in the modern sciences has generated tremendous interest in penalized estimators such as the lasso, scaled lasso, square-root lasso, elastic net, and many others. In this paper, we establish a general oracle inequality for prediction in high-dimensional linear regression with such methods. Since the proof relies only on convexity and continuity arguments, the result holds irrespective of the design matrix and applies to a wide range of penalized estimators. Overall, the bound demonstrates that generic estimators can provide consistent prediction with any design matrix. From a practical point of view, the bound can help to identify the potential of specific estimators, and they can help to get a sense of the prediction accuracy in a given application.

Maximum regularized likelihood estimators (MRLEs) are arguably the most established class of estimators in high-dimensional statistics. In this paper, we derive guarantees for MRLEs in Kullback-Leibler divergence, a general measure of prediction accuracy. We assume only that the densities have a convex parametrization and that the regularization is definite and positive homogenous. The results thus apply to a very large variety of models and estimators, such as tensor regression and graphical models with convex and non-convex regularized methods. A main conclusion is that MRLEs are broadly consistent in prediction - regardless of whether restricted eigenvalues or similar conditions hold.

In applications of graphical models, we typically have more information than just the samples themselves. A prime example is the estimation of brain connectivity networks based on fMRI data, where in addition to the samples themselves, the spatial positions of the measurements are readily available. With particular regard for this application, we are thus interested in ways to incorporate additional knowledge most effectively into graph estimation. Our approach to this is to make neighborhood selection receptive to additional knowledge by strengthening the role of the tuning parameters. We demonstrate that this concept (i) can improve reproducibility, (ii) is computationally convenient and efficient, and (iii) carries a lucid Bayesian interpretation. We specifically show that the approach provides effective estimations of brain connectivity graphs from fMRI data. However, providing a general scheme for the inclusion of additional knowledge, our concept is expected to have applications in a wide range of domains.

Although the Lasso has been extensively studied, the relationship between its prediction performance and the correlations of the covariates is not fully understood. In this paper, we give new insights into this relationship in the context of multiple linear regression. We show, in particular, that the incorporation of a simple correlation measure into the tuning parameter can lead to a nearly optimal prediction performance of the Lasso even for highly correlated covariates. However, we also reveal that for moderately correlated covariates, the prediction performance of the Lasso can be mediocre irrespective of the choice of the tuning parameter. We finally show that our results also lead to near-optimal rates for the least-squares estimator with total variation penalty.

Feature selection is a standard approach to understanding and modeling high-dimensional classification data, but the corresponding statistical methods hinge on tuning parameters that are difficult to calibrate. In particular, existing calibration schemes in the logistic regression framework lack any finite sample guarantees. In this paper, we introduce a novel calibration scheme for penalized logistic regression. It is based on simple tests along the tuning parameter path and satisfies optimal finite sample bounds. It is also amenable to easy and efficient implementations, and it rivals or outmatches existing methods in simulations and real data applications.

The TREX is a recently introduced method for performing sparse high-dimensional regression. Despite its statistical promise as an alternative to the lasso, square-root lasso, and scaled lasso, the TREX is computationally challenging in that it requires solving a non-convex optimization problem. This paper shows a remarkable result: despite the non-convexity of the TREX problem, there exists a polynomial-time algorithm that is guaranteed to find the global minimum. This result adds the TREX to a very short list of non-convex optimization problems that can be globally optimized (principal components analysis being a famous example). After deriving and developing this new approach, we demonstrate that (i) the ability of the preexisting TREX heuristic to reach the global minimum is strongly dependent on the difficulty of the underlying statistical problem, (ii) the new polynomial-time algorithm for TREX permits a novel variable ranking and selection scheme, (iii) this scheme can be incorporated into a rule that controls the false discovery rate (FDR) of included features in the model. To achieve this last aim, we provide an extension of the results of Barber & Candes (2015) to establish that the knockoff filter framework can be applied to the TREX. This investigation thus provides both a rare case study of a heuristic for non-convex optimization and a novel way of exploiting non-convexity for statistical inference.

Lasso is a seminal contribution to high-dimensional statistics, but it hinges on a tuning parameter that is difficult to calibrate in practice. A partial remedy for this problem is Square-Root Lasso, because it inherently calibrates to the noise variance. However, Square-Root Lasso still requires the calibration of a tuning parameter to all other aspects of the model. In this study, we introduce TREX, an alternative to Lasso with an inherent calibration to all aspects of the model. This adaptation to the entire model renders TREX an estimator that does not require any calibration of tuning parameters. We show that TREX can outperform cross-validated Lasso in terms of variable selection and computational efficiency. We also introduce a bootstrapped version of TREX that can further improve variable selection. We illustrate the promising performance of TREX both on synthetic data and on a recent high-dimensional biological data set that considers riboflavin production in B. subtilis.

Lasso is a popular method for high-dimensional variable selection, but it hinges on a tuning parameter that is difficult to calibrate in practice. In this study, we introduce TREX, an alternative to Lasso with an inherent calibration to all aspects of the model. This adaptation to the entire model renders TREX an estimator that does not require any calibration of tuning parameters. We show that TREX can outperform cross-validated Lasso in terms of variable selection and computational efficiency. We also introduce a bootstrapped version of TREX that can further improve variable selection. We illustrate the promising performance of TREX both on synthetic data and on two biological data sets from the fields of genomics and proteomics.

Sparse Filtering is a popular feature learning algorithm for image classification pipelines. In this paper, we connect the performance of Sparse Filtering with spectral properties of the corresponding feature matrices. This connection provides new insights into Sparse Filtering; in particular, it suggests early stopping of Sparse Filtering. We therefore introduce the Optimal Roundness Criterion (ORC), a novel stopping criterion for Sparse Filtering. We show that this stopping criterion is related with pre-processing procedures such as Statistical Whitening and demonstrate that it can make image classification with Sparse Filtering considerably faster and more accurate.