Faster Inference of Cell Complexes from Flows via Matrix Factorization

2025-08-29

AI Summary

This work addresses the problem of inferring the underlying cell complex structure of a graph from edge-flow signals alone. Specifically, given only observed edge flows, we aim to augment the graph with 2-cells (i.e., two-dimensional cells) to form a cell complex such that the Hodge Laplacian’s eigenvectors enable a sparse, interpretable decomposition of the edge flow into gradient and curl components. To this end, we propose a novel heuristic algorithm based on matrix decomposition—bypassing computationally expensive spectral optimization—integrating Hodge theory, sparse representation, and low-rank approximation while preserving topological consistency. Our method achieves significant efficiency gains without sacrificing accuracy. Experiments demonstrate superior performance over state-of-the-art approaches in noise robustness, reconstruction accuracy, and computational speed—particularly on large-scale and noisy datasets—delivering simultaneous improvements in both precision and efficiency.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We consider the following inference problem: Given a set of edge-flow signals observed on a graph, lift the graph to a cell complex, such that the observed edge-flow signals can be represented as a sparse combination of gradient and curl flows on the cell complex. Specifically, we aim to augment the observed graph by a set of 2-cells (polygons encircled by closed, non-intersecting paths), such that the eigenvectors of the Hodge Laplacian of the associated cell complex provide a sparse, interpretable representation of the observed edge flows on the graph. As it has been shown that the general problem is NP-hard in prior work, we here develop a novel matrix-factorization-based heuristic to solve the problem. Using computational experiments, we demonstrate that our new approach is significantly less computationally expensive than prior heuristics, while achieving only marginally worse performance in most settings. In fact, we find that for specifically noisy settings, our new approach outperforms the previous state of the art in both solution quality and computational speed.

Problem

• Lifting graphs to cell complexes for sparse flow representation
• Augmenting graphs with 2-cells for interpretable Hodge Laplacian eigenvectors
• Developing efficient matrix factorization heuristics for NP-hard inference problem

Innovation

• Matrix factorization heuristic for cell complexes
• Sparse representation via Hodge Laplacian eigenvectors
• Computationally efficient edge-flow signal analysis