Statistical and computational tradeoffs in biclustering

We consider the problem of identifying a small sub-matrix of activation in a large noisy matrix. We establish the minimax rate for the problem by showing tight (up to constants) upper and lower bounds on the signal strength needed to identify the sub-matrix. We consider several natural computationally tractable procedures and show that under most parameter scalings they are unable to identify the sub-matrix at the minimax signal strength. While we are unable to directly establish the computational hardness of the problem at the minimax signal strength we discuss connections to some known NP-hard problems and their approximation algorithms.