Inertial, corrected, primal-dual proximal splitting

TL;DR

This paper introduces an inertial primal-dual proximal splitting method with novel correctors, achieving improved convergence rates and acceleration by combining strong convexity and inertial techniques, demonstrated on imaging and inverse problems.

Contribution

It develops a new inertial primal-dual splitting algorithm with correctors for better convergence and acceleration, extending prior methods with gap estimates and strong convexity integration.

Findings

Improved convergence in image processing tasks.

Enhanced performance in sparse Fourier inversion.

Effective acceleration combining strong convexity and inertia.

Abstract

We study inertial versions of primal-dual proximal splitting, also known as the Chambolle--Pock method. Our starting point is the preconditioned proximal point formulation of this method. By adding correctors corresponding to the anti-symmetric part of the relevant monotone operator, using a FISTA-style gap unrolling argument, we are able to derive gap estimates instead of merely ergodic gap estimates. Moreover, based on adding a diagonal component to this corrector, we are able to combine strong convexity based acceleration with inertial acceleration. We test our proposed method on image processing and inverse problems problems, obtaining convergence improvements for sparse Fourier inversion and Positron Emission Tomography.