Rank-based linkage I: triplet comparisons and oriented simplicial complexes

Abstract

Rank-based linkage is a new tool for summarizing a collection $S$ of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on $S$. Rank-based linkage is applied to the $K$-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected $K$-nearest neighbor graph on $S$. In $|S| K^2$ steps it builds an edge-weighted linkage graph $(S, \mathcal{L}, \sigma)$ where $\sigma(\{x, y\})$ is called the in-sway between objects $x$ and $y$. Take $\mathcal{L}_t$ to be the links whose in-sway is at least $t$, and partition $S$ into components of the graph $(S, \mathcal{L}_t)$, for varying $t$. Rank-based linkage is a functor from a category of ``out-ordered'' digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not ``rip apart'' the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Orientation sheaves play in a fundamental role and ensure that partially overlapping data sets can be ``glued'' together. Open combinatorial problems are presented in the last section.