  We consider the problem of sparse normal means estimation in a distributed
setting with communication constraints. We assume there are $M$ machines, each
holding $d$-dimensional observations of a $K$-sparse vector $\mu$ corrupted by
additive Gaussian noise. The $M$ machines are connected in a star topology to a
fusion center, whose 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 at signal-to-noise ratios (SNRs) that are
sufficiently high - but not enough for recovery by any individual machine - the
support of $\mu$ can be correctly recovered with significantly less
communication. Specifically, we present two algorithms for distributed
estimation of a sparse mean vector corrupted by either Gaussian or sub-Gaussian
noise. We then prove that above certain SNR thresholds, with high probability,
these algorithms 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 $KM\ll
\tfrac{d}{\log d}$, 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.
