From Graphs to Topologies: Seeing the Bigger Shape

At the heart of this emerging field is the idea that data often live not just on points or lines, but on topological spaces or shapes. By representing data using mathematical structures called simplicial or cell complexes—generalizations of graphs that can capture triangles, tetrahedra, and higher-dimensional interactions—it is possible to analyze and process signals at multiple levels of connectivity and understand the shape of how information flows.

Tools from algebraic topology— like the Hodge Laplacian make it possible to generalize familiar signal-processing operations—such as filtering, denoising, or spectral analysis—to these richer spaces. 

In simple terms: we can now uncover and reveal global structures and hidden patterns like 

These tools allow signals defined on nodes, edges, and higher-dimensional entities to be processed jointly, revealing global structures like loops, holes, or surfaces that ordinary graph models simply can't see.

Fourier, but for Topologies

When we think of processing signals, the name of Joseph Fourier, the man who taught us how to decompose signals into fundamental frequencies—Fourier Transform, comes naturally. Topological Signal Processing extends that same intuition. By building on the principles of the Fourier transform for time and space, it is possible to generalize the same idea to data defined over graphs or over general arbitrary topologies. This generalization allows connecting methods and insights across different domains and exploring new avenues in a tractable way.

The topological Fourier Transform allows us to "hear" the fundamental oscillating frequencies of signals over complex structures (arbitrary topologies)—from road networks to neural circuits. For instance, in systems with edge flows (like traffic or electrical currents), this approach helps identify circulating versus diverging components—concepts familiar from physics but now applied to abstract data domains. 

While generally abstract, this view not only provides techniques to unravel new insights from data, but also to develop convolutional filtering, sampling, and interpolation techniques, as well as build principled Topological Machine and Deep Learning methods. In the particular case of edge flows, for instance, the topological Fourier transform has intuitive ties with vector calculus and identifies the solenoidal and irrotational components of signals.