Higher-Order Hodge Laplacians for Processing of multi-way Signals

Introduction

Network analysis has revolutionized our understanding of complex systems, and graph-based methods have emerged as powerful tools to process signals on non-Euclidean domains via graph signal processing and graph neural networks.

Importance of Graph Laplacian

The graph Laplacian and related matrices are pivotal to such analyses:

1. The Laplacian serves as an algebraic descriptor of the relationships between nodes. Moreover, it is key for:

   • The analysis of network structure
   • Local operations such as averaging over connected nodes
   • Network dynamics like diffusion and consensus

2. Laplacian eigenvectors are natural basis functions for data on graphs and are endowed with meaningful variability notions for graph signals, akin to Fourier analysis in Euclidean domains.

Limitations of Graphs

However, graphs are ill-equipped to encode multi-way and higher-order relations that are becoming increasingly important to comprehend complex datasets and systems in many applications. Examples include:

• Understanding group dynamics in social systems
• Multi-gene interactions in genetic data
• Multi-way drug interactions

Project Goal

The goal of this project is to develop methods that can utilize such higher-order relations, going from mathematical models to efficient algorithms and software.

Methodological Focus

Specifically, we will focus on ideas from algebraic topology and discrete calculus, according to which the graph Laplacian can be seen as part of a hierarchy of Hodge-Laplacians. These emerge from treating graphs as instances of more general cell complexes that systematically encode couplings between node-tuples of any size.

Ambitions

Our ambition is to:

1. Provide more informative ways to represent and analyze the structure of complex systems, paying special attention to computational efficiency.
2. Translate the success of graph-based signal processing to data on general topological spaces defined by cell complexes.
3. By generalizing from graphs to neural networks on complexes, gain deeper theoretical insights on the principles of graph neural networks as a special case.