  In modern relational machine learning it is common to encounter large graphs
that arise via interactions or similarities between observations in many
domains. Further, in many cases the target entities for analysis are actually
signals on such graphs. We propose to compare and organize such datasets of
graph signals by using an earth mover's distance (EMD) with a geodesic cost
over the underlying graph. Typically, EMD is computed by optimizing over the
cost of transporting one probability distribution to another over an underlying
metric space. However, this is inefficient when computing the EMD between many
signals. Here, we propose an unbalanced graph EMD that efficiently embeds the
unbalanced EMD on an underlying graph into an $L^1$ space, whose metric we call
unbalanced diffusion earth mover's distance (UDEMD). Next, we show how this
gives distances between graph signals that are robust to noise. Finally, we
apply this to organizing patients based on clinical notes, embedding cells
modeled as signals on a gene graph, and organizing genes modeled as signals
over a large cell graph. In each case, we show that UDEMD-based embeddings find
accurate distances that are highly efficient compared to other methods.
