Convergence analysis of equilibrium methods for inverse problems

Solving inverse problems (Ax = y) is central to a variety of practically important fields such as medical imaging, remote sensing, and non-destructive testing. The most successful and theoretically best-understood method is convex variational regularization, where approximate but stable solutions are defined as minimizers of ( |A(·) - yδ|2 / 2 + α R(·)), with (R) a regularization functional. Recent methods such as deep equilibrium models and plug-and-play approaches, however, go beyond variational regularization. Motivated by these innovations, we introduce implicit non-variational (INV) regularization, where approximate solutions are defined as solutions of (A^*(A x - yδ) + α R(x) = 0) for some regularization operator (R). When the regularization operator is the gradient of a functional, INV reduces to classical variational regularization. However, in methods like DEQ and PnP, (R) is not a gradient field, and the existing theoretical foundation remains incomplete. To address this, we establish stability and convergence results in this broader setting, including convergence rates and stability estimates measured via a absolute Bregman distance.