An Optimal Algorithm for Strict Circular Seriation

TL;DR

This paper introduces an optimal algorithm for circular seriation based on strict circular Robinson matrices, extending linear seriation techniques to circular data, and analyzes the statistical properties of the problem.

Contribution

It defines circular Robinson matrices, provides an optimal ${ m O}(n^2)$ algorithm for strict cases, and analyzes the statistical stability of the circular seriation solution.

Findings

Optimal ${ m O}(n^2)$ algorithm for strict circular Robinson matrices.

Establishment of statistical convergence rates in the Kendall-tau metric.

Extension of linear seriation methods to circular data structures.

Abstract

We study the problem of circular seriation, where we are given a matrix of pairwise dissimilarities between $n$ objects, and the goal is to find a {\em circular order} of the objects in a manner that is consistent with their dissimilarity. This problem is a generalization of the classical {\em linear seriation} problem where the goal is to find a {\em linear order}, and for which optimal ${\cal O}(n^2)$ algorithms are known. Our contributions can be summarized as follows. First, we introduce {\em circular Robinson matrices} as the natural class of dissimilarity matrices for the circular seriation problem. Second, for the case of {\em strict circular Robinson dissimilarity matrices} we provide an optimal ${\cal O}(n^2)$ algorithm for the circular seriation problem. Finally, we propose a statistical model to analyze the well-posedness of the circular seriation problem for large $n$. In particular, we establish ${\cal O}(\log(n)/n)$ rates on the distance between any circular ordering found by solving the circular seriation problem to the underlying order of the model, in the Kendall-tau metric.