Point Localization and Density Estimation from Ordinal kNN graphs using Synchronization

TL;DR

This paper introduces a scalable method for embedding unweighted k-nearest neighbor graphs in low-dimensional space using ordinal information and synchronization techniques, also enabling robust density estimation.

Contribution

It proposes a novel local-to-global synchronization approach with scaling for embedding and density recovery from ordinal kNN graphs, improving scalability and robustness.

Findings

Effective embedding of large graphs demonstrated

Robust density estimation via TV-MPLE shown to be feasible

Comparison with LOE highlights advantages in scalability

Abstract

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.