# Incidence Networks for Geometric Deep Learning

## TL;DR

This paper formalizes incidence tensors in geometric deep learning, analyzing their structure and introducing equivariant networks that operate on them, enabling efficient processing of complex structured data.

## Contribution

It introduces a formal framework for incidence tensors, analyzes their structure, and develops equivariant networks with efficient pooling and broadcasting mechanisms.

## Key findings

- Incidence tensors decompose into invariant subsets.
- Decomposition leads to efficient equivariant linear maps.
- Proposed networks handle complex structured data effectively.

## Abstract

Sparse incidence tensors can represent a variety of structured data. For example, we may represent attributed graphs using their node-node, node-edge, or edge-edge incidence matrices. In higher dimensions, incidence tensors can represent simplicial complexes and polytopes. In this paper, we formalize incidence tensors, analyze their structure, and present the family of equivariant networks that operate on them. We show that any incidence tensor decomposes into invariant subsets. This decomposition, in turn, leads to a decomposition of the corresponding equivariant linear maps, for which we prove an efficient pooling-and-broadcasting implementation.