  Given a heterogeneous Gaussian sequence model with unknown mean $\theta \in
\mathbb R^d$ and known covariance matrix $\Sigma =
\operatorname{diag}(\sigma_1^2,\dots, \sigma_d^2)$, we study the signal
detection problem against sparse alternatives, for known sparsity $s$. Namely,
we characterize how large $\epsilon^*>0$ should be, in order to distinguish
with high probability the null hypothesis $\theta=0$ from the alternative
composed of $s$-sparse vectors in $\mathbb R^d$, separated from $0$ in $L^t$
norm ($t \in [1,\infty]$) by at least $\epsilon^*$. We find minimax upper and
lower bounds over the minimax separation radius $\epsilon^*$ and prove that
they are always matching. We also derive the corresponding minimax tests
achieving these bounds. Our results reveal new phase transitions regarding the
behavior of $\epsilon^*$ with respect to the level of sparsity, to the $L^t$
metric, and to the heteroscedasticity profile of $\Sigma$. In the case of the
Euclidean (i.e. $L^2$) separation, we bridge the remaining gaps in the
literature.
