Source Condition Double Robust Inference on Functionals of Inverse Problems

TL;DR

This paper introduces a novel inference method for linear inverse problems that guarantees asymptotic normality under source conditions, leveraging duality and regularization techniques without needing to identify which inverse problem is better posed.

Contribution

It presents the first source condition double robust inference approach for inverse problems, combining primal and dual problem solutions with new guarantees for regularized estimators.

Findings

Ensures asymptotic normality when either primal or dual inverse problem is well-posed.

Develops novel guarantees for iterated Tikhonov regularized estimators.

Applicable over general hypothesis spaces.

Abstract

We consider estimation of parameters defined as linear functionals of solutions to linear inverse problems. Any such parameter admits a doubly robust representation that depends on the solution to a dual linear inverse problem, where the dual solution can be thought as a generalization of the inverse propensity function. We provide the first source condition double robust inference method that ensures asymptotic normality around the parameter of interest as long as either the primal or the dual inverse problem is sufficiently well-posed, without knowledge of which inverse problem is the more well-posed one. Our result is enabled by novel guarantees for iterated Tikhonov regularized adversarial estimators for linear inverse problems, over general hypothesis spaces, which are developments of independent interest.