2023/06/02 by Daniel Obmann, Obmann, Daniel, Hwang, Gyeongha +1
Computer Science · Engineering · Mathematics · #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2306.01421
openalex publication_date 2023/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \( ‖A(⋅) - yδ‖2 / 2 + αR(⋅)\), 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 \(A^*(A x - yδ) + αR(x) = 0\) 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.