Detection boundary in sparse regression

TL;DR

This paper investigates the limits of detecting sparse signals in high-dimensional linear regression with Gaussian noise, establishing the precise conditions under which detection is possible and providing optimal testing procedures.

Contribution

It extends the detection boundary concept from Gaussian sequence models to high-dimensional linear regression, including cases with unknown noise variance.

Findings

Detection boundary characterized for known variance

Detection boundary characterized for unknown variance

Optimal tests achieving the detection boundary provided

Abstract

We study the problem of detection of a p-dimensional sparse vector of parameters in the linear regression model with Gaussian noise. We establish the detection boundary, i.e., the necessary and sufficient conditions for the possibility of successful detection as both the sample size n and the dimension p tend to the infinity. Testing procedures that achieve this boundary are also exhibited. Our results encompass the high-dimensional setting (p>> n). The main message is that, under some conditions, the detection boundary phenomenon that has been proved for the Gaussian sequence model, extends to high-dimensional linear regression. Finally, we establish the detection boundaries when the variance of the noise is unknown. Interestingly, the detection boundaries sometimes depend on the knowledge of the variance in a high-dimensional setting.