Graph Reconstruction by Discrete Morse Theory

TL;DR

This paper provides a theoretical foundation for a discrete Morse theory-based framework to accurately reconstruct hidden graph structures from noisy data, demonstrating correctness under a natural noise model and supporting it with experiments.

Contribution

It introduces a simplified graph reconstruction algorithm based on discrete Morse theory and proves its correctness under a specific noise model.

Findings

The framework can correctly reconstruct the loop structure of the hidden graph.

The reconstructed graph is geometrically close to the ground-truth graph.

Experimental results support the theoretical claims.

Abstract

Recovering hidden graph-like structures from potentially noisy data is a fundamental task in modern data analysis. Recently, a persistence-guided discrete Morse-based framework to extract a geometric graph from low-dimensional data has become popular. However, to date, there is very limited theoretical understanding of this framework in terms of graph reconstruction. This paper makes a first step towards closing this gap. Specifically, first, leveraging existing theoretical understanding of persistence-guided discrete Morse cancellation, we provide a simplified version of the existing discrete Morse-based graph reconstruction algorithm. We then introduce a simple and natural noise model and show that the aforementioned framework can correctly reconstruct a graph under this noise model, in the sense that it has the same loop structure as the hidden ground-truth graph, and is also geometrically close. We also provide some experimental results for our simplified graph-reconstruction algorithm.