Estimating axial symmetry using random projections

TL;DR

This paper investigates methods to identify axes of axial symmetry in multivariate data using random projections, providing theoretical insights and a consistent estimator in two dimensions.

Contribution

It introduces a novel approach using random projections to detect axial symmetry and proves consistency of the estimator in two dimensions.

Findings

Agreement on two random projections suffices in  dimensions

Theoretical relation between symmetry directions and spherical symmetry

Proposed estimator is consistent in the plane

Abstract

This paper studies the problem of identifying directions of axial symmetry in multivariate distributions. Theoretical results are derived on how the measure or cardinality of the set of symmetry directions relates to spherical symmetry. The problem is framed using random projections, leading to a proof that in \(\RR^2\), agreement on two random projections is enough to identify the true axes of symmetry. A corresponding result for higher dimensions is conjectured. An estimator for the symmetry directions is proposed and proved to be consistent in the plane.