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May 2018 On matrix estimation under monotonicity constraints
Sabyasachi Chatterjee, Adityanand Guntuboyina, Bodhisattva Sen
Bernoulli 24(2): 1072-1100 (May 2018). DOI: 10.3150/16-BEJ865

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.

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Sabyasachi Chatterjee. Adityanand Guntuboyina. Bodhisattva Sen. "On matrix estimation under monotonicity constraints." Bernoulli 24 (2) 1072 - 1100, May 2018. https://doi.org/10.3150/16-BEJ865

Information

Received: 1 November 2015; Revised: 1 April 2016; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778359
MathSciNet: MR3706788
Digital Object Identifier: 10.3150/16-BEJ865

Keywords: Adaptation , bivariate isotonic regression , metric entropy bounds , minimax lower bound , Oracle inequalities , tangent cone , variable adaptation

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 2 • May 2018
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