Statistical-Computational Tradeoffs in Planted Problems and Submatrix Localization with a Growing Number of Clusters and Submatrices

We consider two closely related problems: planted clustering and submatrix localization. The planted clustering problem assumes that a random graph is generated based on some underlying clusters of the nodes; the task is to recover these clusters given the graph. The submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real-valued random matrix. Of particular interest is the setting where the number of clusters/submatrices is allowed to grow unbounded with the problem size. These formulations cover several classical models such as planted clique, planted densest subgraph, planted partition, planted coloring, and stochastic block model, which are widely used for studying community detection and clustering/bi-clustering. For both problems, we show that the space of the model parameters (cluster/submatrix size, cluster density, and submatrix mean) can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities: (1) the impossible regime, where all algorithms fail; (2) the hard regime, where the computationally expensive Maximum Likelihood Estimator (MLE) succeeds; (3) the easy regime, where the polynomial-time convexified MLE succeeds; (4) the simple regime, where a simple counting/thresholding procedure succeeds. Moreover, we show that each of these algorithms provably fails in the previous harder regimes. Our theorems establish the minimax recovery limit, which are tight up to constants and hold with a growing number of clusters/submatrices, and provide a stronger performance guarantee than previously known for polynomial-time algorithms. Our study demonstrates the tradeoffs between statistical and computational considerations, and suggests that the minimax recovery limit may not be achievable by polynomial-time algorithms.