从节点数据中学习图的拓扑与限制映射，提升信号处理效率。
Learning Sheaf Laplacian Optimizing Restriction Maps
- 基于层化理论构建可闭式求解的优化框架
- 最小化数据总变差，同时优化各边的限制映射
- 适用于高维子空间数据，对相关性和维度差敏感
本文提出一种新框架，从图节点上的观测数据中推断层化拉普拉斯算子，包括图的拓扑结构和各边的限制映射。该方法基于层化理论，是对图信号处理的重要推广。学习目标是找到使观测数据总变差最小的层化拉普拉斯算子，同时通过优化关联的限制映射实现每条边上的局部变差最小化。相比基于半定规划的替代方法，本方案在数值上更高效，所有基础步骤均可闭式求解。实验在每个节点上定义于不同维度子空间的向量数据上进行，展示了结果图受数据跨相关性和节点间维度差异两个关键因素的影响。
原文摘要 · Abstract (English)
The aim of this paper is to propose a novel framework to infer the sheaf Laplacian, including the topology of a graph and the restriction maps, from a set of data observed over the nodes of a graph. The proposed method is based on sheaf theory, which represents an important generalization of graph signal processing. The learning problem aims to find the sheaf Laplacian that minimizes the total variation of the observed data, where the variation over each edge is also locally minimized by optimizing the associated restriction maps. Compared to alternative methods based on semidefinite programming, our solution is significantly more numerically efficient, as all its fundamental steps are resolved in closed form. The method is numerically tested on data consisting of vectors defined over subspaces of varying dimensions at each node. We demonstrate how the resulting graph is influenced by two key factors: the cross-correlation and the dimensionality difference of the data residing on the graph's nodes.
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