On matrix estimation under monotonicity constraints

By Sabyasachi Chatterjee, Adityanand Guntuboyina, Bodhisattva Sen. 2015

62 citations recorded by the source corpus.

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 i.e., $n^{-1}$ 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 derive, as a consequence, an interesting adaptation property of the LSE which we term variable adaptation -- the LSE performs as well as the oracle estimator when estimating a matrix that is constant along each row/column. Our proofs borrow ideas from empirical process theory and convex geometry and are of independent interest.

Subfields: Computational Mechanics, Computer Networks and Communications, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/37643ea113a80664b24e8b57d5105850199051f8

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In Computational Mechanics

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Citations in the corpus

Optimal rates of statistical seriation (2016, methodology): “The variational formula was first proposed for convex regression [22], and later exploited in several different settings, including matrix estimation with shape constraints [19] and unimodal regression [21].”
Nonparametric Shape-Restricted Regression (2017, background): “In this subsection we briefly review the main results in [29] which considers estimation of a bivariate (d = 2) coordinate-wise nondecreasing regression function.”