  We study the local geometry of testing a mean vector within a
high-dimensional ellipse against a compound alternative. Given samples of a
Gaussian random vector, the goal is to distinguish whether the mean is equal to
a known vector within an ellipse, or equal to some other unknown vector in the
ellipse. Such ellipse testing problems lie at the heart of several
applications, including non-parametric goodness-of-fit testing, signal
detection in cognitive radio, and regression function testing in reproducing
kernel Hilbert spaces. While past work on such problems has focused on the
difficulty in a global sense, we study difficulty in a way that is localized to
each vector within the ellipse. Our main result is to give sharp upper and
lower bounds on the localized minimax testing radius in terms of an explicit
formula involving the Kolmogorov width of the ellipse intersected with a
Euclidean ball. When applied to particular examples, our general theorems yield
interesting rates that were not known before: as a particular case, for testing
in Sobolev ellipses of smoothness $\alpha$, we demonstrate rates that vary from
$(\sigma^2)^{\frac{4 \alpha}{4 \alpha + 1}}$, corresponding to the classical
global rate, to the faster rate $(\sigma^2)^{\frac{8
  \alpha}{8 \alpha + 1}}$, achievable for vectors at favorable locations within
the ellipse. We also show that the optimal test for this problem is achieved by
a linear projection test that is based on an explicit lower-dimensional
projection of the observation vector.
