基于有效集的图全变差的组合重构Optimization
Optimization of Graph Total Variation via Active-Set-based Combinatorial Reconditioning
摘要：
加权图上的结构化凸优化在机器学习和计算机视觉中有许多应用。在这项工作中，我们提出了一种新的自适应预处理策略，在这个问题类的近似算法。我们的预条件子是由一个尖锐的局部线性收敛速率取决于“活动集”在当前的时间间隔的分析。我们表明，嵌套森林分解的不活跃的边缘产生一个有保证的局部线性收敛速率。此外，我们提出了一个实用的贪婪启发式，实现这种嵌套分解，并在几个数值实验表明，我们的修复策略，当应用于近端梯度或原始-对偶混合梯度算法，实现竞争力的表现。我们的研究结果表明，局部收敛分析可以作为一个指导方针，选择可变的度量在邻近算法。
英文摘要：
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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