This paper considers the problem of detecting a few signals in
high-dimensional complex-valued Gaussian data satisfying Johnstone's (2001)
\textit{spiked covariance model}. We focus on the difficult case where signals
are weak in the sense that the sizes of the corresponding covariance spikes are
below the \textit{phase transition threshold} studied in Baik et al (2005). We
derive a simple analytical expression for the maximal possible asymptotic
probability of correct detection holding the asymptotic probability of false
detection fixed. To accomplish this derivation, we establish what we believe to
be a new formula for the \textit{% Harish-Chandra/Itzykson-Zuber (HCIZ)
integral} \int_{\mathcal{U}(p)}e^{\tr(AGBG^{-1})}dG , where A has a
deficient rank r<p. The formula links the HCIZ integral over U(p) to an HCIZ integral over a potentially much smaller unitary group
U(r). We show that the formula generalizes to the integrals over
orthogonal and symplectic groups. In the most general form, it expresses the
hypergeometric function 0F0(α)of two p×p matrix
arguments as a repeated contour integral of the hypergeometric function
0F0(α)of two r×r matrix arguments