  We study the fundamental tradeoffs between statistical accuracy and
computational tractability in the analysis of high dimensional heterogeneous
data. As examples, we study sparse Gaussian mixture model, mixture of sparse
linear regressions, and sparse phase retrieval model. For these models, we
exploit an oracle-based computational model to establish conjecture-free
computationally feasible minimax lower bounds, which quantify the minimum
signal strength required for the existence of any algorithm that is both
computationally tractable and statistically accurate. Our analysis shows that
there exist significant gaps between computationally feasible minimax risks and
classical ones. These gaps quantify the statistical price we must pay to
achieve computational tractability in the presence of data heterogeneity. Our
results cover the problems of detection, estimation, support recovery, and
clustering, and moreover, resolve several conjectures of Azizyan et al. (2013,
2015); Verzelen and Arias-Castro (2017); Cai et al. (2016). Interestingly, our
results reveal a new but counter-intuitive phenomenon in heterogeneous data
analysis that more data might lead to less computation complexity.
