基于几何边界权重的图滤波器，可自动抑制信号能量
Spectral Contraction of Boundary-Weighted Filters on delta-Hyperbolic Graphs
- 用双曲图的Busemann函数定义边权重，反映节点靠近边界程度
- 证明滤波器每轮使信号能量至少衰减固定比例，与曲率相关
- 无需调参、计算轻量，适合处理有层次结构的数据
层级图常呈现树状分支结构，这给传统图滤波器的设计带来挑战。本文引入一种边界加权算子，根据边两端点向图的Gromov边界漂移的距离来重新缩放每条边。利用delta-双曲网络上的Busemann函数，我们证明了该算子谱范数的闭式上界：每个信号在每次传递中都会损失一个由曲率控制的比例能量。结果提供了一种无参数、轻量级的滤波器，其稳定性直接源于几何基本原理，为具有密集或隐藏层次结构的数据提供了新的图信号处理分析工具。
原文摘要 · Abstract (English)
Hierarchical graphs often exhibit tree-like branching patterns, a structural property that challenges the design of traditional graph filters. We introduce a boundary-weighted operator that rescales each edge according to how far its endpoints drift toward the graph's Gromov boundary. Using Busemann functions on delta-hyperbolic networks, we prove a closed-form upper bound on the operator's spectral norm: every signal loses a curvature-controlled fraction of its energy at each pass. The result delivers a parameter-free, lightweight filter whose stability follows directly from geometric first principles, offering a new analytic tool for graph signal processing on data with dense or hidden hierarchical structure.
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