Performing statistical inference in high-dimension is an outstanding challenge. A major source of difficulty is the absence of precise information on the distribution of high-dimensional estimators. Here, we consider linear regression in the high-dimensional regime @math . In this context, we would like to perform inference on a high-dimensional parameters vector @math . Important progress has been achieved in computing confidence intervals for single coordinates @math . A key role in these new methods is played by a certain debiased estimator @math that is constructed from the Lasso. Earlier work establishes that, under suitable assumptions on the design matrix, the coordinates of @math are asymptotically Gaussian provided @math is @math -sparse with @math . The condition @math is stronger than the one for consistent estimation, namely @math . We study Gaussian designs with known or unknown population covariance. When the covariance is known, we prove that the debiased estimator is asymptotically Gaussian under the nearly optimal condition @math . Note that earlier work was limited to @math even for perfectly known covariance. The same conclusion holds if the population covariance is unknown but can be estimated sufficiently well, e.g. under the same sparsity conditions on the inverse covariance as assumed by earlier work. For intermediate regimes, we describe the trade-off between sparsity in the coefficients and in the inverse covariance of the design. We further discuss several applications of our results to high-dimensional inference. In particular, we propose a new estimator that is minimax optimal up to a factor @math for i.i.d. Gaussian designs.
Summary Consider the following three important problems in statistical inference: constructing confidence intervals for the error of a high dimensional (p>n) regression estimator, the linear regression noise level and the genetic signal-to-noise ratio of a continuous-valued trait (related to the heritability). All three problems turn out to be closely related to the little-studied problem of performing inference on the l2-norm of the signal in high dimensional linear regression. We derive a novel procedure for this, which is asymptotically correct when the covariates are multivariate Gaussian and produces valid confidence intervals in finite samples as well. The procedure, called EigenPrism, is computationally fast and makes no assumptions on coefficient sparsity or knowledge of the noise level. We investigate the width of the EigenPrism confidence intervals, including a comparison with a Bayesian setting in which our interval is just 5 wider than the Bayes credible interval. We are then able to unify the three aforementioned problems by showing that EigenPrism with only minor modifications can make important contributions to all three. We also investigate the robustness of coverage and find that the method applies in practice and in finite samples much more widely than just the case of multivariate Gaussian covariates. Finally, we apply EigenPrism to a genetic data set to estimate the genetic signal-to-noise ratio for a number of continuous phenotypes.  In many fields of science, we observe a response variable together with a large number of potential explanatory variables, and would like to be able to discover which variables are truly associated with the response. At the same time, we need to know that the false discovery rate (FDR) - the expected fraction of false discoveries among all discoveries - is not too high, in order to assure the scientist that most of the discoveries are indeed true and replicable. This paper introduces the knockoff filter, a new variable selection procedure controlling the FDR in the statistical linear model whenever there are at least as many observations as variables. This method achieves exact FDR control in finite sample settings no matter the design or covariates, the number of variables in the model, or the amplitudes of the unknown regression coefficients, and does not require any knowledge of the noise level. As the name suggests, the method operates by manufacturing knockoff variables that are cheap - their construction does not require any new data - and are designed to mimic the correlation structure found within the existing variables, in a way that allows for accurate FDR control, beyond what is possible with permutation-based methods. The method of knockoffs is very general and flexible, and can work with a broad class of test statistics. We test the method in combination with statistics from the Lasso for sparse regression, and obtain empirical results showing that the resulting method has far more power than existing selection rules when the proportion of null variables is high. We study the fundamental problems of variance and risk estimation in high dimensional statistical modeling. In particular, we consider the problem of learning a coefficient vector Theta 0 is an element of Rp from noisy linear observations y = X Theta 0 + w is an element of Rn (p > n) and the popular estimation procedure of solving the '1-penalized least squares objective known as the LASSO or Basis Pursuit DeNoising (BPDN). In this context, we develop new estimators for the '2 estimation risk k Theta b- Theta 0k2 and the variance of the noise when distributions of Theta 0 and w are unknown. These can be used to select the regularization parameter optimally. Our approach combines Stein's unbiased risk estimate [Ste81] and the recent results of [BM12a] [BM12b] on the analysis of approximate message passing and the risk of LASSO. We establish high-dimensional consistency of our estimators for sequences of matrices X of increasing dimensions, with independent Gaussian entries. We establish validity for a broader class of Gaussian designs, conditional on a certain conjecture from statistical physics. To the best of our knowledge, this result is the first that provides an asymptotically consistent risk estimator for the LASSO solely based on data. In addition, we demonstrate through simulations that our variance estimation outperforms several existing methods in the literature.