  We consider the problem of sparse normal means estimation in a distributed
setting with communication constraints. We assume there are $M$ machines, each
holding a $d$-dimensional observation of a $K$-sparse vector $\mu$ corrupted by
additive Gaussian noise. A central fusion machine is connected to the $M$
machines in a star topology, and its goal is to estimate the vector $\mu$ with
a low communication budget. Previous works have shown that to achieve the
centralized minimax rate for the $\ell_2$ risk, the total communication must be
high - at least linear in the dimension $d$. This phenomenon occurs, however,
at very weak signals. We show that once the signal-to-noise ratio (SNR) is
slightly higher, the support of $\mu$ can be correctly recovered with much less
communication. Specifically, we present two algorithms for the distributed
sparse normal means problem, and prove that above a certain SNR threshold, with
high probability, they recover the correct support with total communication
that is sublinear in the dimension $d$. Furthermore, the communication
decreases exponentially as a function of signal strength. If in addition $KM\ll
d$, then with an additional round of sublinear communication, our algorithms
achieve the centralized rate for the $\ell_2$ risk. Finally, we present
simulations that illustrate the performance of our algorithms in different
parameter regimes.
