On the geometry of Gaussian Mean Shift Detection

We consider the problem of detecting a shift in the mean of a standard Gaussian random vector, where the direction of the shift belongs to an arbitrary subset T of the Euclidean unit sphere. We introduce a correlation functional $κ(T)$ and establish a general lower bound on the detection radius in terms of this functional. We identify regularity conditions under which this lower bound is optimal up to an absolute constant. As an application, we recover some well-known results in the literature such as the detection radius for s-sparse unit vectors.