Detection boundary in sparse regression

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 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 previously established 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 rate of the detection boundary in high-dimensional setting with unknown variance can be different from the rate for the case of known variance.