On matrix estimation under monotonicity constraints

We consider the problem of estimating an unknown n₁ n₂ matrix ^* from noisy observations under the constraint that ^* 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 multiplicative logarithmic factors, where n=n₁n₂ and that the LSE is minimax rate optimal (up to logarithmic factors). We further prove that for some special ^*, the risk of the LSE could be much smaller than n^-1/2; in fact, it could even be parametric, that is, n^-1 up to logarithmic factors. Such parametric rates occur when the number of “rectangular” blocks of ^* is bounded from above by a constant. We also derive an interesting adaptation property of the LSE which we term variable adaptation – the LSE adapts to the “intrinsic dimension” of the problem and performs as well as the oracle estimator when estimating a matrix that is constant along each row/column. Our proofs, which borrow ideas from empirical process theory, approximation theory and convex geometry, are of independent interest.