  Biclustering structures in data matrices were first formalized in a seminal
paper by John Hartigan (1972) where one seeks to cluster cases and variables
simultaneously. Such structures are also prevalent in block modeling of
networks. In this paper, we develop a unified theory for the estimation and
completion of matrices with biclustering structures, where the data is a
partially observed and noise contaminated data matrix with a certain
biclustering structure. In particular, we show that a constrained least squares
estimator achieves minimax rate-optimal performance in several of the most
important scenarios. To this end, we derive unified high probability upper
bounds for all sub-Gaussian data and also provide matching minimax lower bounds
in both Gaussian and binary cases. Due to the close connection of graphon to
stochastic block models, an immediate consequence of our general results is a
minimax rate-optimal estimator for sparse graphons.
