Adaptive estimation of a function from its Exponential Radon Transform in presence of noise
This work addresses a specific mathematical estimation problem in inverse problems, with incremental improvements in adaptive estimation for ERT data.
The authors tackled the problem of estimating a function from its Exponential Radon Transform (ERT) data without prior smoothness knowledge, proposing a locally adaptive non-parametric kernel estimator that achieves minimax optimal rates up to a log factor for a wide Sobolev regularity class.
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.