Point Localization and Density Estimation from Ordinal Knn Graphs Using Synchronization

We consider the problem of embedding unweighted, directed k-nearest neighbor graphs in low-dimensional Euclidean space. The k-nearest neighbors of each vertex provide ordinal information on the distances between points, but not the distances themselves. Relying only on such ordinal information, along with the low-dimensionality, we 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 localto-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 scale synchronization step. We show our approach compares favorably to the recently proposed Local Ordinal Embedding (LOE) algorithm even in the case of smaller sized problems, and also demonstrate its scalability on large graphs. The above divide-and-conquer paradigm can be of independent interest to the machine learning community when tackling geometric embeddings problems.