Computational Barriers in Minimax Submatrix Detection

This paper studies the minimax detection of a small submatrix of elevated mean in a p× p matrix contaminated by additive Gaussian noise where p tends to infinity. 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 cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size, the following phase transition phenomenon is established: If the submatrix size k = Θ(pα) for any α ∈ (0, 2/3), computational complexity constraints can incur a severe penalty on the statistical performance in the sense that any randomized polynomial time test is minimax suboptimal by a polynomial factor in k; If k = Θ(pα) for any α ∈ (2/3, 1), minimax optimal detection can be attained within constant factors in linear time. Using Schatten norm loss as a representative example, we show that the hardness of attaining the minimax estimation rate can crucially depends on the loss function. Implications on the hardness of support recovery are also discussed.