Graph Mover's Distance: An Efficiently Computable Distance Measure for Geometric Graphs
This work addresses the computational bottleneck in pattern recognition applications that use geometric graphs, offering a more practical distance measure.
The paper tackles the problem of efficiently computing a similarity measure for geometric graphs, which is NP-hard for the existing geometric graph distance (GGD), by proposing the Graph Mover's Distance (GMD) that reduces computation to O(n^3)-time and shows promising results in recognizing letter drawings from the LETTER dataset.
Many applications in pattern recognition represent patterns as a geometric graph. The geometric graph distance (GGD) has recently been studied as a meaningful measure of similarity between two geometric graphs. Since computing the GGD is known to be $\mathcal{NP}$-hard, the distance measure proves an impractical choice for applications. As a computationally tractable alternative, we propose in this paper the Graph Mover's Distance (GMD), which has been formulated as an instance of the earth mover's distance. The computation of the GMD between two geometric graphs with at most $n$ vertices takes only $O(n^3)$-time. Alongside studying the metric properties of the GMD, we investigate the stability of the GGD and GMD. The GMD also demonstrates extremely promising empirical evidence at recognizing letter drawings from the {\tt LETTER} dataset \cite{da_vitoria_lobo_iam_2008}.