  We consider the problem of predictive density estimation under
Kullback-Leibler loss in a high-dimensional Gaussian model with exact sparsity
constraints on the location parameters. We study the first order asymptotic
minimax risk of Bayes predictive density estimates based on product discrete
priors where the proportion of non-zero coordinates converges to zero as
dimension increases. Discrete priors that are product of clustered univariate
priors provide a tractable configuration for diversification of the future risk
and are used for constructing efficient predictive density estimates. We
establish that the Bayes predictive density estimate from an appropriately
designed clustered discrete prior is asymptotically minimax optimal. The
marginals of our proposed prior have infinite clusters of identical sizes. The
within cluster support points are equi-probable and the clusters are
periodically spaced with geometrically decaying probabilities as they move away
from the origin. The cluster periodicity depends on the decay rate of the
cluster probabilities. Under different sparsity regimes, through numerical
experiments, we compare the maximal risk of the Bayes predictive density
estimates from the clustered prior with varied competing estimators including
those based on geometrically decaying non-clustered priors of Johnstone (1994)
and Mukherjee & Johnstone (2017) and obtain encouraging results.
