## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 1072-1100.

### On matrix estimation under monotonicity constraints

Sabyasachi Chatterjee, Adityanand Guntuboyina, and Bodhisattva Sen

#### Abstract

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 multiplicative logarithmic factors, where $n=n_{1}n_{2}$ and that the LSE is minimax rate optimal (up to 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, it could even be parametric, that is, $n^{-1}$ up to logarithmic factors. Such parametric rates occur when the number of “rectangular” blocks of $\mathbf{\theta}^{*}$ 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.

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 1072-1100.

**Dates**

Received: November 2015

Revised: April 2016

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980890

**Digital Object Identifier**

doi:10.3150/16-BEJ865

**Mathematical Reviews number (MathSciNet)**

MR3706788

**Zentralblatt MATH identifier**

06778359

**Keywords**

adaptation bivariate isotonic regression metric entropy bounds minimax lower bound oracle inequalities tangent cone variable adaptation

#### Citation

Chatterjee, Sabyasachi; Guntuboyina, Adityanand; Sen, Bodhisattva. On matrix estimation under monotonicity constraints. Bernoulli 24 (2018), no. 2, 1072--1100. doi:10.3150/16-BEJ865. https://projecteuclid.org/euclid.bj/1505980890