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We derive a second-order ordinary differential equation (ODE), which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show that the continuous time ODE allows for a better understanding of Nesterov's scheme. As a byproduct, we obtain a family(More)
We introduce a new estimator for the vector of coefficients β in the linear model y = Xβ + z, where X has dimensions n × p with p possibly larger than n. SLOPE, short for Sorted LOne Penalized Estimation, is the solution to min b∈R p 1 2 y − Xb 2 2 + λ 1 |b| (1) + λ 2 |b| (2) +. .. + λ p |b| (p) , where λ 1 ≥ λ 2 ≥. .. ≥ λ p ≥ 0 and |b| (1) ≥ |b| (2) ≥. ..(More)
We introduce a novel method for sparse regression and variable selection, which is inspired by modern ideas in multiple testing. Imagine we have observations from the linear model y = Xβ + z, then we suggest estimating the regression coefficients by means of a new estimator called SLOPE, which is the solution to minimize b 1 2 y − Xb 2 2 + λ 1 |b| (1) + λ 2(More)
In this note we give a proof showing that even though the number of false discoveries and the total number of discoveries are not continuous functions of the parameters, the formulas we obtain for the false discovery proportion (FDP) and the power, namely, (B.3) and (B.4) in the paper Statistical Estimation and Testing via the Sorted 1 Norm are(More)
We consider high-dimensional sparse regression problems in which we observe y = Xβ + z, where X is an n × p design matrix and z is an n-dimensional vector of independent Gaussian errors, each with variance σ 2. Our focus is on the recently introduced SLOPE estimator [15], which regularizes the least-squares estimates with the rank-dependent penalty 1≤i≤p λ(More)
The epithelium of the intestinal mucosa is a rapidly self-renewing tissue in the body, and defects in the renewal process occur commonly in various disorders. microRNAs (miRNAs) posttranscriptionally regulate gene expression and are implicated in many aspects of cellular physiology. Here we investigate the role of miRNA-29b (miR-29b) in the regulation of(More)
Sorted LOne Penalized Estimation (SLOPE, [10]) is a relatively new convex optimization procedure which allows for adaptive selection of regressors under sparse high dimensional designs. Here we extend the idea of SLOPE to deal with the situation when one aims at selecting whole groups of explanatory variables instead of single regressors. This approach is(More)