  We consider the problem of embedding unweighted, directed k-nearest neighbor
graphs in low-dimensional Euclidean space. The k-nearest neighbors of each
vertex provides ordinal information on the distances between points, but not
the distances themselves. We use this ordinal information along with the
low-dimensionality to recover the coordinates of the points up to arbitrary
similarity transformations (rigid transformations and scaling). Furthermore, we
also illustrate the possibility of robustly recovering the underlying density
via the Total Variation Maximum Penalized Likelihood Estimation (TV-MPLE)
method. We make existing approaches scalable by using an instance of a
local-to-global algorithm based on group synchronization, recently proposed in
the literature in the context of sensor network localization and structural
biology, which we augment with a scaling synchronization step. We demonstrate
the scalability of our approach on large graphs, and show how it compares to
the Local Ordinal Embedding (LOE) algorithm, which was recently proposed for
recovering the configuration of a cloud of points from pairwise ordinal
comparisons between a sparse set of distances.
