On matrix estimation under monotonicity constraints

We consider the problem of estimating an unknown $n_1 \times n_2$ matrix $\mathbf{\theta^*}$ from noisy observations under the constraint that $\mathbf{\theta}^*$ is nondecreasing in both rows and columns. We consider the least squares estimator (LSE) in this setting and study its risk properties. We show that the worst case risk of the LSE is $n^{-1/2}$, up to logarithmic factors, where $n = n_1 n_2$ and that the LSE is minimax rate optimal upto logarithmic factors. We further prove that for some special $\mathbf{\theta}^*$, the risk of the LSE could be much smaller than $n^{-1/2}$; in fact, could even be parametric i.e., $n^{-1}$ upto logarithmic factors. We derive, as a consequence, an interesting adaptation property of the LSE which we term variable adaptation.