Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs
Asilomar Conference on Signals, Systems and Computers(2023)
摘要
Graph-based signal processing techniques have become essential for handling
data in non-Euclidean spaces. However, there is a growing awareness that these
graph models might need to be expanded into `higher-order' domains to
effectively represent the complex relations found in high-dimensional data.
Such higher-order domains are typically modeled either as hypergraphs, or as
simplicial, cubical or other cell complexes. In this context, cell complexes
are often seen as a subclass of hypergraphs with additional algebraic structure
that can be exploited, e.g., to develop a spectral theory. In this article, we
promote an alternative perspective. We argue that hypergraphs and cell
complexes emphasize different types of relations, which may have
different utility depending on the application context. Whereas hypergraphs are
effective in modeling set-type, multi-body relations between entities, cell
complexes provide an effective means to model hierarchical,
interior-to-boundary type relations. We discuss the relative advantages of
these two choices and elaborate on the previously introduced concept of a
combinatorial complex that enables co-existing set-type and hierarchical
relations. Finally, we provide a brief numerical experiment to demonstrate that
this modelling flexibility can be advantageous in learning tasks.
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