Acceleration of the PDHGM on strongly convex subspaces
This provides improved optimization for image processing problems like denoising and deblurring where full strong convexity is absent, though it appears incremental relative to existing primal-dual methods.
The authors tackled the problem of accelerating primal-dual optimization methods when only partial strong convexity is present, achieving mixed convergence rates of O(1/N^2) for initialization and O(1/N) for dual sequences on image processing tasks.
We propose several variants of the primal-dual method due to Chambolle and Pock. Without requiring full strong convexity of the objective functions, our methods are accelerated on subspaces with strong convexity. This yields mixed rates, $O(1/N^2)$ with respect to initialisation and $O(1/N)$ with respect to the dual sequence, and the residual part of the primal sequence. We demonstrate the efficacy of the proposed methods on image processing problems lacking strong convexity, such as total generalised variation denoising and total variation deblurring.