Topological signal processing handles unoriented higher-order data better
Topological Signal Processing With Unoriented Operators
Machine Learning
Summary
Topological signal processing typically uses orientations to analyze data spread over shapes made of connected points, lines, and triangles. But many signals don't have a clear direction, making traditional tools unreliable. The authors created a new version that works without needing the direction, using a new way to break down signals. This method can better reconstruct data by understanding how signals at different levels combine, especially when signal strengths vary a lot.
What this means in practice
- •For network data analysts: Improve modeling and reconstruction of network flows and interactions without fixed direction assignments in complex data.
- •For sensor network engineers: Enhance processing of signals collected over sensor grids where signals lack a natural orientation, leading to better data recovery.
Authors
Andrea Cavallo, Varun Sarathchandran, Geert Leus, Elvin Isufi
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.