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

Curse of Heterogeneity: Computational Barriers in Sparse Mixture Models and Phase Retrieval