Nuclear-norm-based matrix completion was originally developed for imputing missing entries in low rank, or approximately low rank matrices. However, it has proven widely effective in many problems where there is no reason to assume low-dimensional linear structure in the underlying matrix, as would be imposed by rank constraints. In this manuscript we show that nuclear-norm-based matrix completion attains within a log factor of the minimax rate for estimating the mean structure of matrices that are not necessarily low-rank, but lie in a low-dimensional non-linear manifold, when observations are missing completely at random. In particular, we give upper bounds on the rate of convergence as a function of the number of rows, columns, and observed entries in the matrix, as well as the smoothness and dimension of the non-linear embedding. We additionally give a minimax lower bound: This lower bound agrees with our upper bound (up to a logarithmic factor), which shows that nuclear-norm penalization is (up to log terms) minimax rate optimal for these problems.

In life sciences, high-throughput techniques typically lead to high-dimensional data and often the number of covariates is much larger than the number of observations. This inherently comes with multicollinearity challenging a statistical analysis in a linear regression framework. Penalization methods such as the lasso, ridge regression, the group lasso, and convex combinations thereof, which introduce additional conditions on regression variables, have proven themselves effective. In this study, we introduce a novel approach by combining the lasso and the standardized group lasso leading to meaningful weighting of the predicted ("fitted") outcome which is of primary importance, e.g., in breeding populations. This "fitted" sparse-group lasso was implemented as a proximal-averaged gradient descent method and is part of the R package "seagull" available at CRAN. For the evaluation of the novel method, we executed an extensive simulation study. We simulated genotypes and phenotypes which resemble data of a dairy cattle population. Genotypes at thousands of genomic markers were used as covariates to fit a quantitative response. The proximity of markers on a chromosome determined grouping. In the majority of simulated scenarios, the new method revealed improved prediction abilities compared to other penalization approaches and was able to localize the signals of simulated features.

We present a unified framework for estimation and analysis of generalized additive models in high dimensions. The framework defines a large class of penalized regression estimators, encompassing many existing methods. An efficient computational algorithm for this class is presented that easily scales to thousands of observations and features. We prove minimax optimal convergence bounds for this class under a weak compatibility condition. In addition, we characterize the rate of convergence when this compatibility condition is not met. Finally, we also show that the optimal penalty parameters for structure and sparsity penalties in our framework are linked, allowing cross-validation to be conducted over only a single tuning parameter. We complement our theoretical results with empirical studies comparing some existing methods within this framework.

The white matter contains long-range connections between different brain regions and the organization of these connections holds important implications for brain function in health and disease. Tractometry uses diffusion-weighted magnetic resonance imaging (dMRI) to quantify tissue properties along the trajectories of these connections. Statistical inference from tractometry usually either averages these quantities along the length of each fiber bundle or computes regression models separately for each point along every one of the bundles. These approaches are limited in their sensitivity, in the former case, or in their statistical power, in the latter. We developed a method based on the sparse group lasso (SGL) that takes into account tissue properties along all of the bundles and selects informative features by enforcing both global and bundle-level sparsity. We demonstrate the performance of the method in two settings: i) in a classification setting, patients with amyotrophic lateral sclerosis (ALS) are accurately distinguished from matched controls. Furthermore, SGL identifies the corticospinal tract as important for this classification, correctly finding the parts of the white matter known to be affected by the disease. ii) In a regression setting, SGL accurately predicts "brain age." In this case, the weights are distributed throughout the white matter indicating that many different regions of the white matter change over the lifespan. Thus, SGL leverages the multivariate relationships between diffusion properties in multiple bundles to make accurate phenotypic predictions while simultaneously discovering the most relevant features of the white matter.

Graphical modeling has been broadly useful for exploring the dependence structure among features in a dataset. However, the strength of graphical modeling hinges on our ability to encode and estimate conditional dependencies. In particular, commonly used measures such as partial correlation are only meaningful under strongly parametric (in this case, multivariate Gaussian) assumptions. These assumptions are unverifiable, and there is often little reason to believe they hold in practice. In this paper, we instead consider 3 non-parametric measures of conditional dependence. These measures are meaningful without structural assumptions on the multivariate distribution of the data. In addition, we show that for 2 of these measures there are simple, strong plug-in estimators that require only the estimation of a conditional mean. These plug-in estimators (1) are asymptotically linear and non-parametrically efficient, (2) allow incorporation of flexible machine learning techniques for conditional mean estimation, and (3) enable the construction of valid Wald-type confidence intervals. In addition, by leveraging the influence function of these estimators, one can obtain intervals with simultaneous coverage guarantees for all pairs of features.

Abstract Background Statistical analyses of biological problems in life sciences often lead to high-dimensional linear models. To solve the corresponding system of equations, penalization approaches are often the methods of choice. They are especially useful in case of multicollinearity, which appears if the number of explanatory variables exceeds the number of observations or for some biological reason. Then, the model goodness of fit is penalized by some suitable function of interest. Prominent examples are the lasso, group lasso and sparse-group lasso. Here, we offer a fast and numerically cheap implementation of these operators via proximal gradient descent. The grid search for the penalty parameter is realized by warm starts. The step size between consecutive iterations is determined with backtracking line search. Finally, seagull -the R package presented here- produces complete regularization paths. Results Publicly available high-dimensional methylation data are used to compare seagull to the established R package SGL . The results of both packages enabled a precise prediction of biological age from DNA methylation status. But even though the results of seagull and SGL were very similar ( R 2 > 0.99), seagull computed the solution in a fraction of the time needed by SGL . Additionally, seagull enables the incorporation of weights for each penalized feature. Conclusions The following operators for linear regression models are available in seagull : lasso, group lasso, sparse-group lasso and Integrative LASSO with Penalty Factors (IPF-lasso). Thus, seagull is a convenient envelope of lasso variants.

The broadly neutralizing antibody (bnAb) VRC01 is being evaluated for its efficacy to prevent HIV-1 infection in the Antibody Mediated Prevention (AMP) trials. A secondary objective of AMP utilizes sieve analysis to investigate how VRC01 prevention efficacy (PE) varies with HIV-1 envelope (Env) amino acid (AA) sequence features. An exhaustive analysis that tests how PE depends on every AA feature with sufficient variation would have low statistical power. To design an adequately powered primary sieve analysis for AMP, we modeled VRC01 neutralization as a function of Env AA sequence features of 611 HIV-1 gp160 pseudoviruses from the CATNAP database, with objectives: (1) to develop models that best predict the neutralization readouts; and (2) to rank AA features by their predictive importance with classification and regression methods. The dataset was split in half, and machine learning algorithms were applied to each half, each analyzed separately using cross-validation and hold-out validation. We selected Super Learner, a nonparametric ensemble-based cross-validated learning method, for advancement to the primary sieve analysis. This method predicted the dichotomous resistance outcome of whether the IC50 neutralization titer of VRC01 for a given Env pseudovirus is right-censored (indicating resistance) with an average validated AUC of 0.868 across the two hold-out datasets. Quantitative log IC50 was predicted with an average validated R2 of 0.355. Features predicting neutralization sensitivity or resistance included 26 surface-accessible residues in the VRC01 and CD4 binding footprints, the length of gp120, the length of Env, the number of cysteines in gp120, the number of cysteines in Env, and 4 potential N-linked glycosylation sites; the top features will be advanced to the primary sieve analysis. This modeling framework may also inform the study of VRC01 in the treatment of HIV-infected persons.

In predictive modeling applications, it is often of interest to determine the relative contribution of subsets of features in explaining the variability of an outcome. It is useful to consider this variable importance as a function of the unknown, underlying data-generating mechanism rather than the specific predictive algorithm used to fit the data. In this paper, we connect these ideas in nonparametric variable importance to machine learning, and provide a method for efficient estimation of variable importance when building a predictive model using a neural network. We show how a single augmented neural network with multi-task learning simultaneously estimates the importance of many feature subsets, improving on previous procedures for estimating importance. We demonstrate on simulated data that our method is both accurate and computationally efficient, and apply our method to both a study of heart disease and for predicting mortality in ICU patients.

We present a novel approach for nonparametric regression using wavelet basis functions. Our proposal, waveMesh, can be applied to non-equispaced data with sample size not necessarily a power of 2. We develop an efficient proximal gradient descent algorithm for computing the estimator and establish adaptive minimax convergence rates. The main appeal of our approach is that it naturally extends to additive and sparse additive models for a potentially large number of covariates. We prove minimax optimal convergence rates under a weak compatibility condition for sparse additive models. The compatibility condition holds when we have a small number of covariates. Additionally, we establish convergence rates for when the condition is not met. We complement our theoretical results with empirical studies comparing waveMesh to existing methods.

We consider the problem of predicting an outcome variable on the basis of a small number of covariates, using an interpretable yet non-additive model. We propose convex regression with interpretable sharp partitions (CRISP) for this task. CRISP partitions the covariate space into blocks in a data- adaptive way, and fits a mean model within each block. Unlike other partitioning methods, CRISP is fit using a non-greedy approach by solving a convex optimization problem, resulting in low- variance fits. We explore the properties of CRISP, and evaluate its performance in a simulation study and on a housing price data set.