Finiteness for the k-factor model and chirality varieties

TL;DR

This paper proves that for fixed k and increasing n, the algebraic varieties from the k-factor model in statistics and chirality varieties in chemistry are finitely characterisable by polynomial equations, aiding in their analysis.

Contribution

It establishes finite characterisation of polynomial equations for two classes of algebraic varieties arising from applications, when k is fixed and n grows.

Findings

Polynomial equations finitely characterise the varieties for fixed k and large n.

Results apply to both statistical covariance models and chemical chirality measurements.

Provides a foundation for testing membership in these varieties using polynomial equations.

Abstract

This paper deals with two families of algebraic varieties arising from applications. First, the k-factor model in statistics, consisting of n-times-n covariance matrices of n observed Gaussian variables that are pairwise independent given k hidden Gaussian variables. Second, chirality varieties inspired by applications in chemistry. A point in such a chirality variety records chirality measurements of all k-subsets among an n-set of ligands. Both classes of varieties are given by a parameterisation, while for applications having polynomial equations would be desirable. For instance, such equations could be used to test whether a given point lies in the variety. We prove that in a precise sense, which is different for the two classes of varieties, these equations are finitely characterisable when k is fixed and n grows.