Source Condition Double Robust Inference on Functionals of Inverse Problems
This provides a robust statistical inference framework for inverse problems in fields like causal inference and econometrics, though it appears incremental as it builds on existing doubly robust concepts.
The paper tackles the problem of estimating parameters defined as linear functionals of solutions to linear inverse problems by introducing a source condition double robust inference method that ensures asymptotic normality as long as either the primal or dual inverse problem is sufficiently well-posed, without requiring knowledge of which is more well-posed.
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.