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We consider the problem of embedding unweighted, directed k-nearest neighborgraphs in low-dimensional Euclidean space. The k-nearest neighbors of eachvertex provides ordinal information on the distances between points, but notthe distances themselves. We use this ordinal information along with thelow-dimensionality to recover the coordinates of the points up to arbitrarysimilarity transformations (rigid transformations and scaling). Furthermore, wealso illustrate the possibility of robustly recovering the underlying densityvia the Total Variation Maximum Penalized Likelihood Estimation (TV-MPLE)method. We make existing approaches scalable by using an instance of alocal-to-global algorithm based on group synchronization, recently proposed inthe literature in the context of sensor network localization and structuralbiology, which we augment with a scaling synchronization step. We demonstratethe scalability of our approach on large graphs, and show how it compares tothe Local Ordinal Embedding (LOE) algorithm, which was recently proposed forrecovering the configuration of a cloud of points from pairwise ordinalcomparisons between a sparse set of distances.

Point Localization and Density Estimation from Ordinal kNN graphs using Synchronization