Inverting the Fisher information operator in non-linear models

Abstract: We consider non-linear regression models corrupted by generic noise when the regression functions form a non-linear subspace of L^2, relevant in non-linear PDE inverse problems and data assimilation. We show that when the score of the model is injective, the Fisher information operator is automatically invertible between well-identified Hilbert spaces, and we provide an operational characterization of these spaces. This allows us to construct in broad generality the efficient Gaussian involved in the classical minimax and convolution theorems to establish information lower bounds, that are typically achieved by Bayesian algorithms thus showing optimality of these methods. We illustrate our results on time-evolution PDE models for reaction-diffusion and Navier-Stokes equations.