Estimation of convex supports from noisy measurements

TL;DR

This paper introduces a new method for estimating convex supports from noisy data contaminated with Gaussian noise, achieving near-optimal convergence rates without tuning parameters.

Contribution

The paper presents a parameter-free estimator for convex support estimation under Gaussian noise, with provable convergence rates and computational efficiency.

Findings

Estimator converges at rate O_d(log log n / sqrt(log n)) in Hausdorff distance.

Computational complexity is (O(log n))^{(d-1)/2}.

Lower bound for minimax rate is Omega_d(1 / log^2 n).

Abstract

A popular class of problem in statistics deals with estimating the support of a density from $n$ observations drawn at random from a $d$-dimensional distribution. The one-dimensional case reduces to estimating the end points of a univariate density. In practice, an experimenter may only have access to a noisy version of the original data. Therefore, a more realistic model allows for the observations to be contaminated with additive noise. In this paper, we consider estimation of convex bodies when the additive noise is distributed according to a multivariate Gaussian distribution, even though our techniques could easily be adapted to other noise distributions. Unlike standard methods in deconvolution that are implemented by thresholding a kernel density estimate, our method avoids tuning parameters and Fourier transforms altogether. We show that our estimator, computable in $(O(\ln n))^{(d-1)/2}$ time, converges at a rate of $ O_d(\log\log n/\sqrt{\log n}) $ in Hausdorff distance, in accordance with the polylogarithmic rates encountered in Gaussian deconvolution problems. Part of our analysis also involves the optimality of the proposed estimator. We provide a lower bound for the minimax rate of estimation in Hausdorff distance that is $\Omega_d(1/\log^2 n)$.