Statistical-Computational Phase Transitions in Planted Models: The High-Dimensional Setting

The planted models assume that a graph is generated from some unknown 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, planted densest subgraph and planted coloring. Of particular interest is the high-dimensional setting where the number of clusters is allowed to grow with the number of nodes. We show that the space of model parameters can be partitioned into four disjoint regions: (1) the impossible regime, where all algorithms fail; (2) the hard regime, where the computationally intractable Maximum Likelihood Estimator (MLE) succeeds, and no polynomial-time method is known; (3) the easy regime, where the polynomial-time convexified MLE succeeds; (4) the simple regime, where a simple counting/thresholding procedure succeeds. Moreover, each of these algorithms provably fails in the previous harder regimes. Our theorems establish the first minimax recovery results for the high-dimensional setting, and provide the best known guarantees for polynomialtime algorithms. These results demonstrate the tradeoffs between statistical and computational considerations.