We devise a communication-efficient approach to distributed sparse regression in the high-dimensional setting. The key idea is to average debiased or desparsified lasso estimators. We show the approach converges at the same rate as the lasso as long as the dataset is not split across too many machines, and consistently estimates the support under weaker conditions than the lasso. On the computational side, we propose a new parallel and computationally-efficient algorithm to compute the approximate inverse covariance required in the debiasing approach, when the dataset is split across samples. We further extend the approach to generalized linear models.

Biclustering (also known as submatrix localization) is a problem of high practical relevance in exploratory analysis of high-dimensional data. We develop a framework for performing statistical inference on biclusters found by score-based algorithms. Since the bicluster was selected in a data dependent manner by a biclustering or localization algorithm, this is a form of selective inference. Our framework gives exact (non-asymptotic) confidence intervals and p-values for the significance of the selected biclusters. Further, we generalize our approach to obtain exact inference for Gaussian statistics.

Penalized M-estimators are used in diverse areas of science and engineering to fit high-dimensional models with some low-dimensional structure. Often, the penalties are \emph{geometrically decomposable}, \ie\ can be expressed as a sum of (convex) support functions. We generalize the notion of irrepresentable to geometrically decomposable penalties and develop a general framework for establishing consistency and model selection consistency of M-estimators with such penalties. We then use this framework to derive results for some special cases of interest in bioinformatics and statistical learning.