Adaptive estimation of a function from its Exponential Radon Transform in presence of noise

TL;DR

This paper introduces a locally adaptive non-parametric estimator for functions from their Exponential Radon Transform data, achieving near-minimax rates without prior smoothness knowledge, but with limitations on optimality under pointwise risk.

Contribution

It develops a new adaptive kernel estimator for ERT data that is nearly minimax optimal across Sobolev classes without prior smoothness assumptions.

Findings

Estimator achieves near-minimax rates up to a log factor.

No fully optimal adaptive estimator exists on Sobolev scale under pointwise risk.

Proposed method adapts to unknown smoothness of functions.

Abstract

In this article we propose a locally adaptive strategy for estimating a function from its Exponential Radon Transform (ERT) data, without prior knowledge of the smoothness of functions that are to be estimated. We build a non-parametric kernel type estimator and show that for a class of functions comprising a wide Sobolev regularity scale, our proposed strategy follows the minimax optimal rate up to a $\log{n}$ factor. We also show that there does not exist an optimal adaptive estimator on the Sobolev scale when the pointwise risk is used and in fact the rate achieved by the proposed estimator is the adaptive rate of convergence.