— Ch. 1 · Foundations And Motivation —
Topological deep learning.
~5 min read · Ch. 1 of 6
Traditional deep learning models like convolutional neural networks excel at processing data on regular grids. These systems assume datasets reside in highly structured Euclidean spaces. Images serve as the primary example where pixel values sit on a fixed grid structure. Scientific and real-world data often defy this rigid organization. Point clouds, meshes, time series, and scalar fields exhibit more intricate domains. Graphs represent another common non-Euclidean structure encountered in computations. The field of geometric deep learning originally proposed signal-processing perspectives for such data types. Early work focused heavily on graphs where connectivity relies on nodes and edges. Subsequent research extended these concepts to simplicial complexes and CW complexes. Topological data analysis offered an independent perspective by describing structural information as shape. This framework inherently recognizes multiple scales ranging from local details to global structures. Initial methods worked only with smaller datasets before new descriptors emerged. Modern researchers now integrate topological information into existing deep-learning models or train directly on topological domains.
Topological Domains Defined
A finite set S of abstract entities forms the basis for defining neighborhoods through functions. Edges provide one method to define relations among entities within that set. Standard graph edges model binary relations connecting typically two entities. Many applications require relations incorporating more than just two entities simultaneously. Higher-order relations allow broader neighborhood functions to capture multi-way interactions. Simplicial complexes represent the simplest form of higher-order domains available today. These structures admit hierarchical relationships making them suitable for various complex applications. Hodge theory can be naturally defined upon simplicial complexes without extra effort. Cell complexes generalize simplicial complexes while providing greater flexibility in defining relations. Each cell in a cell complex remains homeomorphic to an open ball attached via maps. Boundary cells of each component are also recognized as valid cells within the complex. Hypergraphs allow arbitrary set-type relations among entities without imposed constraints. Relations in hypergraphs do not adhere to constraints found in other models. Combinatorial complexes bridge gaps between simplicial complexes, cell complexes, and hypergraphs. They combine features from all three while offering additional modeling flexibility. Hierarchical structures and set-type relations emerge as key properties across these domains. A rank function attaches non-negative integer values to preserve set inclusion order.