Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An optimization parameter for seriation of noisy data

Jeannette Janssen, Mahya Ghandehari

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, $Γ_\max$, which measures how badly a given matrix fails to be Robinson similarity. Namely, a matrix is Robinson similarity precisely when its $Γ_\max$ attains zero, and a matrix with small $Γ_\max$ is close (in the normalized $\ell^1$-norm) to a Robinson similarity matrix. Moreover, both $Γ_\max$ 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.

An optimization parameter for seriation of noisy data

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, , which measures how badly a given matrix fails to be Robinson similarity. Namely, a matrix is Robinson similarity precisely when its attains zero, and a matrix with small is close (in the normalized -norm) to a Robinson similarity matrix. Moreover, both 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.

Paper Structure

This paper contains 6 sections, 11 theorems, 52 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Definitions and main results
  3. Robinson similarity approximation for binary matrices
  4. Robinson similarity approximations of general matrices
  5. Improvement through preprocessing
  6. Conclusions and further work

Key Result

Theorem \oldthetheorem

\newlabelthm:main0 For every $A\in{\mathcal{A}}_n$, there exists a Robinson matrix $R\in {\mathcal{A}}_n$ so that Moreover, $R$ can be computed in polynomial time. In addition, if $A$ is binary, then there exists a binary matrix $R$ satisfying the conditions of the theorem.

Figures (2)

  • Figure 1: The black region is convex around the diagonal.
  • Figure 2: Regions ${\rm UR} (a,b)$ (blue) and ${\rm LL} (a,b)$ (red)

Theorems & Definitions (26)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Proof 1
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Proof 2
  • Proof 3: Proof of Claim 1
  • Proof 4: Proof of Claim 2
  • Lemma \oldthetheorem
  • ...and 16 more