Deep Model-Based Architectures for Inverse Problems under Mismatched Priors

TL;DR

This paper investigates deep model-based architectures for inverse problems, providing new theoretical error bounds and numerical insights into their performance when the CNN priors are mismatched due to distribution shifts or approximations.

Contribution

It offers the first theoretical analysis and explicit error bounds for DMBAs under mismatched CNN priors, addressing a key gap in existing research.

Findings

Explicit error bounds derived for mismatched CNN priors

Numerical results show performance degradation under distribution shifts

Analysis includes approximate statistical estimators like MAP and MMSE

Abstract

There is a growing interest in deep model-based architectures (DMBAs) for solving imaging inverse problems by combining physical measurement models and learned image priors specified using convolutional neural nets (CNNs). For example, well-known frameworks for systematically designing DMBAs include plug-and-play priors (PnP), deep unfolding (DU), and deep equilibrium models (DEQ). While the empirical performance and theoretical properties of DMBAs have been widely investigated, the existing work in the area has primarily focused on their performance when the desired image prior is known exactly. This work addresses the gap in the prior work by providing new theoretical and numerical insights into DMBAs under mismatched CNN priors. Mismatched priors arise naturally when there is a distribution shift between training and testing data, for example, due to test images being from a different distribution than images used for training the CNN prior. They also arise when the CNN prior used for inference is an approximation of some desired statistical estimator (MAP or MMSE). Our theoretical analysis provides explicit error bounds on the solution due to the mismatched CNN priors under a set of clearly specified assumptions. Our numerical results compare the empirical performance of DMBAs under realistic distribution shifts and approximate statistical estimators.