Lecture Notes on High Dimensional Linear Regression

Comment: Fisher Lecture: Dimension Reduction in Regression [arXiv:0708.3774]

These lecture notes consist of three chapters. In the first chapter we present oracle inequalities for the prediction error of the Lasso and square-root Lasso and briefly describe the scaled Lasso. In the second chapter we establish asymptotic linearity of a de-sparsified Lasso. This implies asymptotic normality under certain conditions and therefore can be used to construct confidence intervals for parameters of interest. A similar line of reasoning can be invoked to derive ...

Traditional statistical inference considers relatively small data sets and the corresponding theoretical analysis focuses on the asymptotic behavior of a statistical estimator when the number of samples approaches infinity. However, many data sets encountered in modern applications have dimensionality significantly larger than the number of training data available, and for such problems the classical statistical tools become inadequate. In order to analyze high-dimensional da...

Sparse linear regression is a vast field and there are many different algorithms available to build models. Two new papers published in Statistical Science study the comparative performance of several sparse regression methodologies, including the lasso and subset selection. Comprehensive empirical analyses allow the researchers to demonstrate the relative merits of each estimator and provide guidance to practitioners. In this discussion, we summarize and compare the two stud...

Because of the advance in technologies, modern statistical studies often encounter linear models with the number of explanatory variables much larger than the sample size. Estimation and variable selection in these high-dimensional problems with deterministic design points is very different from those in the case of random covariates, due to the identifiability of the high-dimensional regression parameter vector. We show that a reasonable approach is to focus on the projectio...

We develop theoretical results that establish a connection across various regression methods such as the non-negative least squares, bounded variable least squares, simplex constrained least squares, and lasso. In particular, we show in general that a polyhedron constrained least squares problem admits a locally unique sparse solution in high dimensions. We demonstrate the power of our result by concretely quantifying the sparsity level for the aforementioned methods. Further...

This paper develops an approach to inference in a linear regression model when the number of potential explanatory variables is larger than the sample size. The approach treats each regression coefficient in turn as the interest parameter, the remaining coefficients being nuisance parameters, and seeks an optimal interest-respecting transformation, inducing sparsity on the relevant blocks of the notional Fisher information matrix. The induced sparsity is exploited through a m...

We compare the risk of ridge regression to a simple variant of ordinary least squares, in which one simply projects the data onto a finite dimensional subspace (as specified by a Principal Component Analysis) and then performs an ordinary (un-regularized) least squares regression in this subspace. This note shows that the risk of this ordinary least squares method is within a constant factor (namely 4) of the risk of ridge regression.

It is clear that conventional statistical inference protocols need to be revised to deal correctly with the high-dimensional data that are now common. Most recent studies aimed at achieving this revision rely on powerful approximation techniques, that call for rigorous results against which they can be tested. In this context, the simplest case of high-dimensional linear regression has acquired significant new relevance and attention. In this paper we use the statistical phys...

I developed the lecture notes based on my ``Linear Model'' course at the University of California Berkeley over the past seven years. This book provides an intermediate-level introduction to the linear model. It balances rigorous proofs and heuristic arguments. This book provides R code to replicate all simulation studies and case studies.