# Faster Inference of Cell Complexes from Flows via Matrix Factorization

## TL;DR

This paper introduces a fast, matrix-factorization heuristic for inferring cell complexes from flow data on graphs, improving efficiency and performance in noisy scenarios compared to previous methods.

## Contribution

It proposes a novel matrix-factorization-based heuristic to efficiently infer cell complexes from flow data, addressing NP-hardness and outperforming prior heuristics in speed and accuracy.

## Key findings

- Significantly reduces computational time compared to prior heuristics.
- Achieves comparable solution quality in most settings.
- Outperforms previous methods in noisy data scenarios.

## 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.