Combinatorial Preconditioners for Proximal Algorithms on Graphs

Thomas Möllenhoff, Zhenzhang Ye, Tao Wu, Daniel Cremers
Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, PMLR 84:38-47, 2018.

Abstract

We present a novel preconditioning technique for proximal optimization methods that relies on graph algorithms to construct effective preconditioners. Such combinatorial preconditioners arise from partitioning the graph into forests. We prove that certain decompositions lead to a theoretically optimal condition number. We also show how ideal decompositions can be realized using matroid partitioning and propose efficient greedy variants thereof for large-scale problems. Coupled with specialized solvers for the resulting scaled proximal subproblems, the preconditioned algorithm achieves competitive performance in machine learning and vision applications.

Cite this Paper

BibTeX
@InProceedings{pmlr-v84-mollenhoff18a, title = {Combinatorial Preconditioners for Proximal Algorithms on Graphs}, author = {Möllenhoff, Thomas and Ye, Zhenzhang and Wu, Tao and Cremers, Daniel}, booktitle = {Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics}, pages = {38--47}, year = {2018}, editor = {Storkey, Amos and Perez-Cruz, Fernando}, volume = {84}, series = {Proceedings of Machine Learning Research}, month = {09--11 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v84/mollenhoff18a/mollenhoff18a.pdf}, url = {https://proceedings.mlr.press/v84/mollenhoff18a.html}, abstract = {We present a novel preconditioning technique for proximal optimization methods that relies on graph algorithms to construct effective preconditioners. Such combinatorial preconditioners arise from partitioning the graph into forests. We prove that certain decompositions lead to a theoretically optimal condition number. We also show how ideal decompositions can be realized using matroid partitioning and propose efficient greedy variants thereof for large-scale problems. Coupled with specialized solvers for the resulting scaled proximal subproblems, the preconditioned algorithm achieves competitive performance in machine learning and vision applications.} }
Endnote
%0 Conference Paper %T Combinatorial Preconditioners for Proximal Algorithms on Graphs %A Thomas Möllenhoff %A Zhenzhang Ye %A Tao Wu %A Daniel Cremers %B Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2018 %E Amos Storkey %E Fernando Perez-Cruz %F pmlr-v84-mollenhoff18a %I PMLR %P 38--47 %U https://proceedings.mlr.press/v84/mollenhoff18a.html %V 84 %X We present a novel preconditioning technique for proximal optimization methods that relies on graph algorithms to construct effective preconditioners. Such combinatorial preconditioners arise from partitioning the graph into forests. We prove that certain decompositions lead to a theoretically optimal condition number. We also show how ideal decompositions can be realized using matroid partitioning and propose efficient greedy variants thereof for large-scale problems. Coupled with specialized solvers for the resulting scaled proximal subproblems, the preconditioned algorithm achieves competitive performance in machine learning and vision applications.
APA
Möllenhoff, T., Ye, Z., Wu, T. & Cremers, D.. (2018). Combinatorial Preconditioners for Proximal Algorithms on Graphs. Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 84:38-47 Available from https://proceedings.mlr.press/v84/mollenhoff18a.html.

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