  Mixture models are a standard approach to dealing with heterogeneous data
with non-i.i.d. structure. However, when the dimension $p$ is large relative to
sample size $n$ and where either or both of means and covariances/graphical
models may differ between the latent groups, mixture models face statistical
and computational difficulties and currently available methods cannot
realistically go beyond $p \! \sim \! 10^4$ or so. We propose an approach
called Model-based Clustering via Adaptive Projections (MCAP). Instead of
estimating mixtures in the original space, we work with a low-dimensional
representation obtained by linear projection. The projection dimension itself
plays an important role and governs a type of bias-variance tradeoff with
respect to recovery of the relevant signals. MCAP sets the projection dimension
automatically in a data-adaptive manner, using a proxy for the assignment risk.
Combining a full covariance formulation with the adaptive projection allows
detection of both mean and covariance signals in very high dimensional
problems. We show real-data examples in which covariance signals are reliably
detected in problems with $p \! \sim \! 10^4$ or more, and simulations going up
to $p = 10^6$. In some examples, MCAP performs well even when the mean signal
is entirely removed, leaving differential covariance structure in the
high-dimensional space as the only signal. Across a number of regimes, MCAP
performs as well or better than a range of existing methods, including a
recently-proposed $\ell_1$-penalized approach; and performance remains broadly
stable with increasing dimension. MCAP can be run "out of the box" and is fast
enough for interactive use on large-$p$ problems using standard desktop
computing resources.
