Computational Barriers in Minimax Submatrix Detection

This paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive Gaussian noise. To investigate the tradeoff between statistical performance and computational cost from a complexity theoretic perspective, we consider a sequence of discretized models which are asymptotically equivalent in the sense of Le Cam. Under the hypothesis that the Planted Clique detection problem is computationally intractable when the clique size is of smaller order than the square root of the graph size, the following phase transition phenomenon is established: There exists a critical value of the submatrix size, below which there is a gap between the minimax detection rate and what can be achieved by computationally efficient procedures, and above which there exists no such gap. The result suggests that computational complexity constraint can incur a severe penalty on the statistical efficiency in the highly sparse regime.