Lens Depth Function and k-Relative Neighborhood Graph: Versatile Tools for Ordinal Data Analysis

In recent years it has become popular to study machine learning problems based on ordinal distance information rather than numerical distance measurements. By ordinal distance information we refer to binary answers to distance comparisons such as d(A,B) < d(C,D). For many problems in machine learning and statistics it is unclear how to solve them in such a scenario. Up to now, the main approach is to explicitly construct an ordinal embedding of the data points in the Euclidean space, an approach that has a number of drawbacks. In this paper, we propose algorithms for the problems of medoid estimation, outlier identification, classification and clustering when given only ordinal distance comparisons. They are based on estimating the lens depth function and the k-relative neighborhood graph on a data set. Our algorithms are simple, can easily be parallelized, avoid many of the drawbacks of an ordinal embedding approach and are much faster.