Feasibility-based fixed point networks

Howard Heaton,Aviv Gibali,Samy Wu Fung,Wotao Yin

doi:10.1186/s13663-021-00706-3

Abstract

Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable recovery with respect to data perturbations. Hand-chosen analytic regularization can yield desirable theoretical guarantees, but such approaches have limited effectiveness recovering signals due to their inability to leverage large amounts of available data. To this end, this work fuses data-driven regularization and convex feasibility in a theoretically sound manner. This is accomplished using feasibility-based fixed point networks (F-FPNs). Each F-FPN defines a collection of nonexpansive operators, each of which is the composition of a projection-based operator and a data-driven regularization operator. Fixed point iteration is used to compute fixed points of these operators, and weights of the operators are tuned so that the fixed points closely represent available data. Numerical examples demonstrate performance increases by F-FPNs when compared to standard TV-based recovery methods for CT reconstruction and a comparable neural network based on algorithm unrolling. Codes are available on Github: github.com/howardheaton/feasibility_fixed_point_networks.

Highlights

Inverse problems arise in numerous applications such as medical imaging [1,2,3,4], phase retrieval [5,6,7], geophysics [8,9,10,11,12,13], and machine learning [14,15,16,17,18]
5.3 Experiment results Our results show that feasibility-based fixed point networks (F-fixed point network (FPN)) outperforms all classical methods as well as the unrolled data-driven method
We emphasize that the type of noise depends on each individual ray in a similar manner to [120], making the measurements more noisy than some related works. This noise and ill-posedness of our underdetermined setup are illustrated by the poor quality of analytic method reconstructions. nearly identical in structure to F-FPNs, these results show the unrolled method to be inferior to F-FPNs in these experiments

Summary

Introduction

Inverse problems arise in numerous applications such as medical imaging [1,2,3,4], phase retrieval [5,6,7], geophysics [8,9,10,11,12,13], and machine learning [14,15,16,17,18]. The goal of inverse problems is to recover a signal ud from a collection of indirect noisy measurements d These quantities are typically related by a linear mapping A via d = Aud + ε,. An ideal regularizer will leverage available data to best capture the core properties that should be exhibited by output reconstruction estimates of true signals. Contribution The core contribution of this work is to connect powerful feasibilityseeking algorithms to data-driven regularization in a manner that maintains theoretical guarantees. This is accomplished by presenting a feasibility-based fixed point network (FFPN) framework that solves a learned feasibility problem. Numerical examples are provided that demonstrate notable performance benefits to our proposed formulation when compared to TV-based methods and fixed-depth neural networks formed by algorithm unrolling. Numerical examples are provided with discussion and conclusions (Sections 5 and 6)

Feasibility problem

Data-driven feasibility problem

Conclusion

Method