An optimization parameter for seriation of noisy data

TL;DR

This paper introduces a new polynomial-time computable parameter, $\Gamma_ ext{max}$, that quantifies how close a matrix is to being a Robinson similarity matrix, aiding in the seriation of noisy data.

Contribution

The paper defines $\Gamma_ ext{max}$ as a novel measure for assessing and approximating Robinson similarity matrices in noisy conditions.

Findings

$\Gamma_ ext{max}$$ is zero for perfect Robinson matrices.

Small $\Gamma_ ext{max}$$ indicates proximity to Robinson similarity.

Both $\Gamma_ ext{max}$$ and its approximation can be computed efficiently.

Abstract

A square symmetric matrix is a Robinson similarity matrix if entries in its rows and columns are non-decreasing when moving towards the diagonal. A Robinson similarity matrix can be viewed as the affinity matrix between objects arranged in linear order, where objects closer together have higher affinity. We define a new parameter, $\Gamma_\max$, which measures how badly a given matrix fails to be Robinson similarity. Namely, a matrix is Robinson similarity precisely when its $\Gamma_\max$ attains zero, and a matrix with small $\Gamma_\max$ is close (in the normalized $\ell^1$-norm) to a Robinson similarity matrix. Moreover, both $\Gamma_\max$ and the Robinson similarity approximation can be computed in polynomial time. Thus, our parameter recognizes Robinson similarity matrices which are perturbed by noise, and can therefore be a useful tool in the problem of seriation of noisy data.