# Preconditioning of a Generalized Forward-Backward Splitting and Application to Optimization on Graphs

arXiv: https://arxiv.org/abs/1504.07699 · HTML: https://mlsanity.com/paper/1504.07699/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2015.

## Built on by (most cited first, 2 of 2)

- [Is there anything left? Measuring semantic residuals of objects removed from 3D Gaussian Splatting](https://mlsanity.com/paper/2503.17574/) (2025)
- [Variational models for signal processing with Graph Neural Networks](https://mlsanity.com/paper/2103.16337/) (2021)

## Abstract (arXiv)

We present a preconditioning of a generalized forward-backward splitting algorithm for finding a zero of a sum of maximally monotone operators sumi=1n Ai + B with B cocoercive, involving only the computation of B and of the resolvent of each Ai separately. This allows in particular to minimize functionals of the form sumi=1n gi + f with f smooth, using only the computation of the gradient of f and of the proximity operator of each gi separately. By adapting the underlying metric, such preconditioning can serve two practical purposes: first, it might accelerate the convergence, or second, it might simplify the computation of the resolvent of Ai for some i. In addition, in many cases of interest, our preconditioning strategy allows the economy of storage and computation concerning some auxiliary variables. In particular, we show how this approach can handle large-scale, nonsmooth, convex optimization problems structured on graphs, which arises in many image processing or learning applications, and that it compares favorably to alternatives in the literature.