On Unifying Randomized Methods For Inverse Problems

TL;DR

This paper unifies the analysis of randomized methods for inverse problems within a stochastic optimization framework, establishing convergence guarantees and enabling the discovery of new methods.

Contribution

It provides a unified theoretical analysis of various randomized inverse problem methods, proving convergence and error bounds, and introduces new randomization techniques.

Findings

Proves asymptotic convergence for multiple randomized methods.

Establishes a non-asymptotic error bound applicable to these methods.

Numerical results verify theoretical convergence and explore different rates.

Abstract

This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a sample average approximation. More importantly, we are able to prove a single theoretical result that guarantees the asymptotic convergence for a variety of randomized methods. Additionally, viewing randomized methods as a sample average approximation enables us to prove, for the first time, a single non-asymptotic error result that holds for randomized methods under consideration. Another important consequence of our unified framework is that it allows us to discover new randomization methods. We present various numerical results for linear, nonlinear, algebraic, and PDE-constrained inverse problems that verify the theoretical convergence results and provide a discussion on the apparently different convergence rates and the behavior for various randomized methods.