Hunt For The Unique, Stable, Sparse And Fast Feature Learning On Graphs
For the purpose of learning on graphs, we hunt for a graph feature representation that exhibit certain uniqueness, stability and sparsity properties while also being amenable to fast computation. This leads to the discovery of family of graph spectral distances (denoted as FGSD) and their based graph feature representations, which we prove to possess most of these desired properties. To both evaluate the quality of graph features produced by FGSD and demonstrate their utility, we apply them to the graph classification problem. Through extensive experiments, we show that a simple SVM based classification algorithm, driven with our powerful FGSD based graph features, significantly outperforms all the more sophisticated state-of-art algorithms on the unlabeled node datasets in terms of both accuracy and speed; it also yields very competitive results on the labeled datasets -despite the fact it does not utilize any node label information.
The authors propose a kernel for unlabeled graphs based on the histogram of the pairwise node distances where the notion of distance is defined via the Eigen decomposition of the laplacian matrix. They show that such a notion enjoys useful uniqueness and stability properties, and that when viewed as an embedding it is a generalization of known graph embedding and dimensional reduction approaches. 
The work would benefit from a clearer exposition (briefly anticipating the notion of "distance" or of invariance, uniqueness, stability and sparsity albeit in an informal way). 
Figure 3 needs to be rethought as it is not informative in the current form (perhaps showing the histogram of the number of non zero elements in the two cases?)
The authors should be careful not to oversell the results of Theorem 1, as in practice it is not known how does the number of non unique harmonic spectra increases with the increase in graph size (after all graphs in real world cases have sizes well exceeding 10 nodes). Also it is not clear whether having near identical spectra could be compatible with large structural changes as measured by other sensible graph metrics.
Theorem 5 in the current form is not informative as it encapsulates in \theta a variability that is hard for the reader to gauge.
The stability analysis seems to be based on the condition that the variation of weights be positive; perhaps the authors want to add a brief comment on what could be expected if the variation is unrestricted? 
An empirical analysis of the effect of the histogram bin size should probably be included. 
The work of Costa and De Grave. "Fast neighborhood subgraph pairwise distance kernel." ICML 2010, seems to be related as they use the histogram of (shortest path) distances between (subgraphs centered around) each pair of nodes, and should probably be referenced and/or compared with. 
There are a few typos and badly formatted references.  
However the work is of interest and of potential impact.