On Inverse Problems, Parameter Estimation, and Domain Generalization

Signal restoration and inverse problems are key elements in most real-world data science applications. In the past decades, with the emergence of machine learning methods, inversion of measurements has become a popular step in almost all physical applications, which is normally executed prior to downstream tasks that often involve parameter estimation. In this work, we analyze the general problem of parameter estimation in an inverse problem setting. First, we address the domain-shift problem by re-formulating it in direct relation with the discrete parameter estimation analysis. We analyze a significant vulnerability in current attempts to enforce domain generalization, which we dubbed the Double Meaning Theorem. Our theoretical findings are experimentally illustrated for domain shift examples in image deblurring and speckle suppression in medical imaging. We then proceed to a theoretical analysis of parameter estimation given observed measurements before and after data processing involving an inversion of the observations. We compare this setting for invertible and non-invertible (degradation) processes. We distinguish between continuous and discrete parameter estimation, corresponding with regression and classification problems, respectively. Our theoretical findings align with the well-known information-theoretic data processing inequality, and to a certain degree question the common misconception that data-processing for inversion, based on modern generative models that may often produce outstanding perceptual quality, will necessarily improve the following parameter estimation objective. It is our hope that this paper will provide practitioners with deeper insights that may be leveraged in the future for the development of more efficient and informed strategic system planning, critical in safety-sensitive applications.