FGOT: Graph Distances based on Filters and Optimal Transport
This work addresses the challenge of comparing and aligning graphs without known correspondences, which is important for applications in network analysis and machine learning, though it appears to be an incremental improvement combining existing techniques.
The authors tackled the problem of graph comparison and alignment by introducing a filter graph distance based on optimal transport and filtered graph signals, which offers flexibility in prioritizing different spectral information. Their experiments showed significant performance improvements in graph alignment and classification tasks while achieving practical speed through approximation methods.
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.