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Mixture models are a standard approach to dealing with heterogeneous datawith non-i.i.d. structure. However, when the dimension $p$ is large relative tosample size $n$ and where either or both of means and covariances/graphicalmodels may differ between the latent groups, mixture models face statisticaland computational difficulties and currently available methods cannotrealistically go beyond $p \! \sim \! 10^4$ or so. We propose an approachcalled Model-based Clustering via Adaptive Projections (MCAP). Instead ofestimating mixtures in the original space, we work with a low-dimensionalrepresentation obtained by linear projection. The projection dimension itselfplays an important role and governs a type of bias-variance tradeoff withrespect to recovery of the relevant signals. MCAP sets the projection dimensionautomatically in a data-adaptive manner, using a proxy for the assignment risk.Combining a full covariance formulation with the adaptive projection allowsdetection of both mean and covariance signals in very high dimensionalproblems. We show real-data examples in which covariance signals are reliablydetected in problems with $p \! \sim \! 10^4$ or more, and simulations going upto $p = 10^6$. In some examples, MCAP performs well even when the mean signalis entirely removed, leaving differential covariance structure in thehigh-dimensional space as the only signal. Across a number of regimes, MCAPperforms as well or better than a range of existing methods, including arecently-proposed $\ell_1$-penalized approach; and performance remains broadlystable with increasing dimension. MCAP can be run "out of the box" and is fastenough for interactive use on large-$p$ problems using standard desktopcomputing resources.

Model-based clustering in very high dimensions via adaptive projections