Optimization of Graph Total Variation via Active-Set-based Combinatorial Reconditioning

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Zhenzhang Ye, Thomas Möllenhoff, Tao Wu, Daniel Cremers ;
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:657-668, 2020.

Abstract

Structured convex optimization on weighted graphs finds numerous applications in machine learning and computer vision. In this work, we propose a novel adaptive preconditioning strategy for proximal algorithms on this problem class. Our preconditioner is driven by a sharp analysis of the local linear convergence rate depending on the "active set" at the current iterate. We show that nested-forest decomposition of the inactive edges yields a guaranteed local linear convergence rate. Further, we propose a practical greedy heuristic which realizes such nested decompositions and show in several numerical experiments that our reconditioning strategy, when applied to proximal gradient or primal-dual hybrid gradient algorithm, achieves competitive performances. Our results suggest that local convergence analysis can serve as a guideline for selecting variable metrics in proximal algorithms.

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