一种用于超图p-拉普拉斯算子的算子分裂算法及其在缺失数据恢复中的应用
An operator-splitting algorithm for the hypergraph $p$-Laplacian with applications to missing data recovery
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中文总结 AI 辅助
本文针对超图p-拉普拉斯正则化，提出算子分裂算法，通过高斯-赛德尔方式处理超边，利用辅助变量和ADMM克服非光滑性，证明算法收敛性，并通过缺失数据恢复问题测试，表明其比现有方法更快。
中文摘要 AI 辅助
超图p-拉普拉斯正则化是数据分析中的一个基本模型，在各种任务中有成功应用。它旨在最小化一个非光滑且通常大规模的目标函数，该函数定义为超边上Lipschitz正则化的p次幂之和。本文提出一种用于超图p-拉普拉斯算子的算子分裂算法，能以高斯-赛德尔方式分别处理超边。每个子问题可视为超边上的广义图Lipschitz学习，为此引入辅助变量克服非光滑性并通过一步乘子交替方向法（ADMM）求解。所得算法在超边上顺序执行近端ADMM更新，并证明了其收敛性。通过在缺失数据恢复问题（如图像稀疏修复和半监督学习）上测试该算法，表明它比现有方法更快。
英文摘要
Hypergraph $p$-Laplacian regularization is a fundamental model in data analysis with successful applications in various tasks. It aims to minimize a nonsmooth and typically large-scale objective function defined as the sum of the $p$-th powers of the Lipschitz regularization over hyperedges. In this paper, we propose an operator-splitting algorithm for the hypergraph $p$-Laplacian that allows us to handle hyperedges separately in a Gauss-Seidel fashion. Each subproblem can be viewed as a generalized graph Lipschitz learning on a hyperedge, for which we introduce an auxiliary variable to overcome the nonsmoothness and solve it with one step of the alternating direction method of multipliers (ADMM). The resulting algorithm performs proximal ADMM updates sequentially over the hyperedges, and its convergence is proven. We test the algorithm on missing data recovery problems, including image sparse inpainting and semi-supervised learning, to demonstrate that it is faster than existing methods.
