  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.
