拓扑信号处理与非定向算子

中文摘要 AI 辅助

拓扑信号处理（TSP）利用定向边界算子处理单纯复形上的信号，这对于流信号或当拓扑不变量在任务中发挥作用时是自然的选择。然而，许多高阶信号并不具有方向性，对它们应用定向算子是不明确的，因为这引入了单纯形方向的任意选择。我们研究了一种非定向TSP（UTSP）框架，该框架用非定向关联矩阵替代定向边界。首先，我们证明了任意单纯形层级之间的非定向关联矩阵和拉普拉斯矩阵具有类似图的谱性质。其次，由于去除方向性会移除霍奇分解，我们引入了一个非定向的对应物，称为交互阶分解，它量化了高阶信号中有多少可以通过聚合低阶信号来解释。第三，我们利用这种分解推导出用于信号重建的正则化器，该正则化器分别惩罚每个交互阶。在真实世界数据上的实验表明，阶感知正则化器优于定向基线，当信号能量在各阶之间分布不均匀时，其增益最大。

英文摘要

Topological signal processing (TSP) processes signals on simplicial complexes with oriented boundary operators, which is the natural choice for flow signals or when the topological invariants play a role for the task at hand. However, many higher-order signals carry no orientation, and applying oriented operators to them is not well-defined since it introduces an arbitrary choice of simplex orientation. We study an unoriented TSP (UTSP) framework that replaces oriented boundaries with unoriented incidence matrices. First, we show that unoriented incidence and Laplacian matrices between arbitrary simplicial levels admit graph-like spectral properties. Second, since dropping orientation removes the Hodge decomposition, we introduce an unoriented counterpart, termed interaction-order decomposition, which quantifies how much of a higher-order signal is explained by aggregating lower-order signals. Third, we use this decomposition to derive regularizers for signal reconstruction that penalize each interaction order separately. Experiments on real-world data show that the order-aware regularizers outperform oriented baselines, with the largest gains when the signal energy is unevenly distributed across orders.

发表机构

• Delft University of Technology（代尔夫特理工大学）

机构由 AI 辅助整理，请以论文原文为准。