Spectral analysis of 2D outlier layout

TL;DR

This paper introduces a spectral analysis method for 2D outlier detection in point distributions, utilizing operator theory and a Hessenberg matrix-based algorithm that is robust to certain perturbations.

Contribution

It combines advanced operator theory with a novel algorithm for separating outliers in 2D point clouds, providing exact moment transformation formulas.

Findings

Exact formulas for transforming point distribution moments

Algorithm's robustness to trace class and Hilbert-Schmidt class perturbations

Effective outlier separation in general 2D point distributions

Abstract

Thompson's partition of a cyclic subnormal operator into normal and completely non-normal components is combined with a non-commutative calculus for hyponormal operators for separating outliers from the cloud, in rather general point distributions in the plane. The main result provides exact transformation formulas from the power moments of the prescribed point distribution into the moments of the uniform mass carried by the cloud. The proposed algorithm solely depends on the Hessenberg matrix associated to the original data. The robustness of the algorithm is reflected by the insensitivity of the output under trace class, or by a theorem of Voiculescu, under certain Hilbert-Schmidt class, additive perturbations of the Hessenberg matrix.