[Paper Review] The convex class of realisable unit covariances

Raphaël Lachièze-Rey|arXiv (Cornell University)|Jan 18, 2013

Point processes and geometric inequalities12 references3 citations

TL;DR

This paper characterizes the convex polytope of realizable unit covariances for finite discrete random sets, proving that Matheron's conjecture—asserting a specific combinatorial condition suffices for realizability—is false. It establishes that the set of realizable unit covariances forms a convex polytope, and shows that all $k$-tuples of extreme points form singular covariance matrices when $k < N$, implying their convex hulls lie on the boundary of the polytope for $N$-element sets.

ABSTRACT

This paper concerns the characterisation of second order marginals for random sets in a discrete setting. Under the instance of unit covariances, this problem possesses a combinatorial symmetry, exploited jointly in the companion paper to give a heuristic procedure to check realisability. In particular we disprove Matheron's conjecture, and explicit partially the structure of the convex body formed by realisable unit covariances in a finite set.

Motivation & Objective

• To characterize the set of second-order marginals (unit covariances) realizable by random binary fields on a finite set.
• To resolve the long-standing realisability problem for unit covariances in discrete random set models.
• To disprove Matheron’s conjecture that a specific combinatorial condition fully characterizes realizable unit covariances.
• To describe the geometric structure of the convex body of realizable unit covariances using tools from convex geometry and polyhedral combinatorics.

Proposed method

• The paper models the set of realizable unit covariances as a convex polytope $\mathscr{U}_{\mathcal{X}}^*$ in $\mathbb{R}^{\mathcal{X} \times \mathcal{X}}$, leveraging the fact that unit fields induce symmetric, positive semi-definite covariance matrices with unit diagonal.
• It uses the necessary and sufficient condition that a symmetric function $\rho$ is realizable if and only if $\sum_{x,y} \alpha_{x,y} \rho_{x,y} \geq \varkappa_\alpha$ for all $\alpha \in \mathscr{F}_{\mathcal{X}}'$, where $\varkappa_\alpha = \inf_{{\mathsf{u}} \in \mathsf{B}_{\mathcal{X}}} \sum_{x,y} \alpha_{x,y} \mathsf{u}_x \mathsf{u}_y$.
• The study employs convex geometry techniques: extreme points (vertices), faces, and the $k$-th order hypergraph structure of the polytope to analyze its boundary and internal structure.
• It proves that for any $k < N$, the convex hull of any $k$-tuple of extreme points (corresponding to deterministic $\{-1,1\}^N$ vectors) lies on the boundary of $\mathscr{U}_N^*$, due to the induced covariance matrix being singular.
• The analysis relies on the fact that if $k < N$, there exists a non-trivial linear dependence among the $k$ vectors, forcing the corresponding random field to satisfy a.s. $\sum_j \lambda_j X_j = 0$, implying singularity of the covariance matrix.
• The paper uses the duality between $V$-description (vertices) and $H$-description (half-spaces) to describe the polytope, and derives structural results on its boundary via affine hyperplane intersections.

Experimental results

Research questions

• RQ1 Is Matheron’s conjecture—that a specific set of combinatorial inequalities fully characterizes realizable unit covariances—valid for all finite sets?
• RQ2 What is the complete geometric structure of the convex body $\mathscr{U}_N^*$ of realizable unit covariances for $N$-element sets?
• RQ3 For which $k$-tuples of extreme points does the convex hull lie on the boundary of $\mathscr{U}_N^*$?
• RQ4 Can the realizability of a given symmetric function $\rho$ with unit diagonal be determined via a finite system of linear inequalities?
• RQ5 How do singularities in the covariance matrix relate to the combinatorial structure of the underlying random field?

Key findings

• The paper disproves Matheron’s conjecture by showing that the combinatorial conditions he proposed are necessary but not sufficient for realizability in general.
• The set of realizable unit covariances forms a convex polytope $\mathscr{U}_N^*$, which is fully characterized by its vertices corresponding to deterministic $\{-1,1\}^N$-valued fields.
• For any $k < N$, the convex hull of any $k$-tuple of vertices lies entirely on the boundary of $\mathscr{U}_N^*$, due to the induced covariance matrix being singular.
• The $k$-th order hypergraph structure of $\mathscr{U}_N^*$ is complete: every $k$-tuple of extreme points spans a simplex contained in the boundary of the polytope.
• The boundary structure of $\mathscr{U}_N^*$ is governed by linear dependence: if $k < N$, any random field supported on $k$ deterministic vectors satisfies a non-trivial linear constraint a.s., forcing the covariance matrix to be singular.
• The paper provides a theoretical foundation for a heuristic algorithm to test realizability, based on convex geometry and the finite description of the polytope’s facets.