Representing Edge Flows on Graphs via Sparse Cell Complexes

TL;DR

This paper extends the Hodge decomposition approach to cellular complexes for representing edge flows on graphs, introducing a new flow representation learning problem and an efficient approximation algorithm that outperforms existing methods.

Contribution

It generalizes the Hodge-based flow representation to cellular complexes and proposes a novel, NP-hard problem with an efficient approximation algorithm.

Findings

The proposed algorithm achieves lower approximation error than state-of-the-art methods.

The method is computationally efficient on real-world and synthetic data.

The approach provides sparse, interpretable representations of edge flows.

Abstract

Obtaining sparse, interpretable representations of observable data is crucial in many machine learning and signal processing tasks. For data representing flows along the edges of a graph, an intuitively interpretable way to obtain such representations is to lift the graph structure to a simplicial complex: The eigenvectors of the associated Hodge-Laplacian, respectively the incidence matrices of the corresponding simplicial complex then induce a Hodge decomposition, which can be used to represent the observed data in terms of gradient, curl, and harmonic flows. In this paper, we generalize this approach to cellular complexes and introduce the flow representation learning problem, i.e., the problem of augmenting the observed graph by a set of cells, such that the eigenvectors of the associated Hodge Laplacian provide a sparse, interpretable representation of the observed edge flows on the graph. We show that this problem is NP-hard and introduce an efficient approximation algorithm for its solution. Experiments on real-world and synthetic data demonstrate that our algorithm outperforms state-of-the-art methods with respect to approximation error, while being computationally efficient.