FGOT: Graph Distances based on Filters and Optimal Transport

TL;DR

This paper introduces the filter graph distance, an optimal transport-based metric for graph comparison that leverages filtered graph signals, enabling flexible, efficient, and accurate graph alignment and classification.

Contribution

The work presents a novel filter graph distance based on optimal transport, along with an approximate cost function and a stochastic mirror gradient descent algorithm for efficient graph alignment.

Findings

Significant improvement in graph alignment accuracy.

Enhanced graph classification performance.

Framework is practical due to computational efficiency.

Abstract

Graph comparison deals with identifying similarities and dissimilarities between graphs. A major obstacle is the unknown alignment of graphs, as well as the lack of accurate and inexpensive comparison metrics. In this work we introduce the filter graph distance. It is an optimal transport based distance which drives graph comparison through the probability distribution of filtered graph signals. This creates a highly flexible distance, capable of prioritising different spectral information in observed graphs, offering a wide range of choices for a comparison metric. We tackle the problem of graph alignment by computing graph permutations that minimise our new filter distances, which implicitly solves the graph comparison problem. We then propose a new approximate cost function that circumvents many computational difficulties inherent to graph comparison and permits the exploitation of fast algorithms such as mirror gradient descent, without grossly sacrificing the performance. We finally propose a novel algorithm derived from a stochastic version of mirror gradient descent, which accommodates the non-convexity of the alignment problem, offering a good trade-off between performance accuracy and speed. The experiments on graph alignment and classification show that the flexibility gained through filter graph distances can have a significant impact on performance, while the difference in speed offered by the approximation cost makes the framework applicable in practical settings.