  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.
