Statistical-Computational Tradeoffs in Planted Models: The High-Dimensional Setting

The planted models assume that a graph is generated from a set of clusters by randomly placing edges between nodes according to their cluster memberships; the task is to recover the clusters given the graph. Special cases include planted clique, planted partition and planted coloring. This paper studies the statisticalcomputational tradeoffs of these models. Our focus is the high-dimensional setting, where the number of clusters is allowed to grow with the number of nodes. We show that the complexities of cluster recovery exhibit phase transitions. In particular, the space of model parameters can be partitioned into four regions with decreasing statistical and computational complexities: (1) the impossible regime, where all algorithms fail; (2) the hard regime, where the exponential-time Maximum Likelihood Estimator (MLE) succeeds; (3) the easy regime, where a polynomial-time convexified MLE succeeds; (4) the simple regime, where a simple algorithm based on counting degrees and common neighbors succeeds. Moreover, each of these algorithms is likely to fail in the harder regime.