Learning on Simplicial Complexes
From Convolutions to Generative Models
Maosheng Yang (TU Delft - Multimedia Computing)
G Leus – Promotor (TU Delft - Signal Processing Systems)
E. Isufi – Copromotor (TU Delft - Multimedia Computing)
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Abstract
Machine learning has been growing beyond data living on Euclidean spaces (e.g., texts, audios, images). Graph machine learning models, e.g., graph neural networks (GNNs), succeed in learning from graph-structured data using the graph topological information. In this thesis, we focus on a new domain, simplicial complexes. Not only are they a popular higher-order network model that generalizes graph models encoding pairwise relations between nodes, but they also allow us to support signals on various network objects (nodes, edges, and faces). For example, edge flows defined on a simplicial complex can be studied in terms of both divergent and rotational properties, providing a better model for real-world flows like traffic flows, water flows, money flows, etc.
The main theme of the thesis is to develop principled machine learning models for signals on simplicial complexes. By principled, we mean that the models leverage the intrinsic priors of the domain and the signals, namely, the topological structure of simplicial compelxes and the Hodge decomposition of simplicial signals. The latter states that, for example, edge flows can be decomposed into a divergence-free part and a curl-free part, each modeling the distinct properties of real-world flows — the conservation of flows (e.g., water flows) and the rotational properties of flows (e.g., electric currents)....