  A key technique of machine learning and computer vision is to embed discrete
weighted graphs into continuous spaces for further downstream processing.
Embedding discrete hierarchical structures in hyperbolic geometry has proven
very successful since it was shown that any weighted tree can be embedded in
that geometry with arbitrary low distortion. Various optimization methods for
hyperbolic embeddings based on common models of hyperbolic geometry have been
studied. In this paper, we consider Hilbert geometry for the standard simplex
which is isometric to a vector space equipped with the variation polytope norm.
We study the representation power of this Hilbert simplex geometry by embedding
distance matrices of graphs. Our findings demonstrate that Hilbert simplex
geometry is competitive to alternative geometries such as the Poincar\'e
hyperbolic ball or the Euclidean geometry for embedding tasks while being fast
and numerically robust.
