Topological Signal Processing With Unoriented Operators

2026-09-21

AI Summary

研究无向拓扑信号处理(UTSP)框架，使用无向关联矩阵替代有向边界算子，并提出交互阶分解方法以改善信号重构。

Abstract

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.

Problem

• Topological Signal Processing
• Unoriented Operators
• Simplicial Complexes
• Higher-order Signals

Innovation

• Unoriented Topological Signal Processing
• Interaction-Order Decomposition
• Spectral Properties
• Regularizers for Signal Reconstruction