Estimation of Monge Matrices

Abstract

Monge matrices and their permuted versions known as pre-Monge matrices naturally appear in many domains across science and engineering. While the rich structural properties of such matrices have long been leveraged for algorithmic purposes, little is known about their impact on statistical estimation. In this work, we propose to view this structure as a shape constraint and study the problem of estimating a Monge matrix subject to additive random noise. More specifically, we establish the minimax rates of estimation of Monge and pre-Monge matrices. In the case of pre-Monge matrices, the minimax-optimal leastsquares estimator is not efficiently computable, and we propose two efficient estimators and establish their rates of convergence. Our theoretical findings are supported by numerical experiments.

Optimal Rates for Estimation of Two-Dimensional Totally Positive Distributions

“…As we have discussed above, MTP 2 is also called log-supermodular. In the recent paper [19], we studied estimation of supermodular matrices (also known as anti-Monge matrices) under sub-Gaussian noise. We note that the proof techniques used in [19] are the starting point for the proofs in this paper, but are extended to the context density estimation.…”

“…In the recent paper [19], we studied estimation of supermodular matrices (also known as anti-Monge matrices) under sub-Gaussian noise. We note that the proof techniques used in [19] are the starting point for the proofs in this paper, but are extended to the context density estimation. In a parallel work [12], the authors study a related but slightly different model under Gaussian noise, and their proof techniques could potentially be extended to yield rates similar to the ones found in this paper.…”

“…However, since the number of constraints is of the order n 1 n 2 , solving the linear systems in each iteration step of these solvers can take a long time without specialized solvers. We address this issue by employing a proximal Newton method, whose main step consists in a projection onto the set of constraints, which in turn can be solved by a variant of Dykstra's algorithm, as discussed in [19]. In this section, in order to emphasize the connection to computing projections, we think about (2.1) as a minimization problem instead of a maximization problem by changing the sign in front of the objective.…”

“…Second, problem (4) can be efficiently solved by a variant of Dykstra's algorithm, as shown in [19]. The idea is to split up the projection onto M ∩ C(Y ) into the projection onto C(Y ) and the sets…”

“…The main task in the sequel is to bound the right-hand side of (6.1.1). The strategy builds upon a spectral decomposition technique from the paper [19] on Monge matrix estimation.…”