A perturbative approach to the reconstruction of the eigenvalue spectrum of a normal covariance matrix from a spherically truncated counterpart

TL;DR

This paper introduces a perturbative method to reconstruct the original covariance matrix of a multinormal distribution from its spherically truncated version, providing an analytical approach and comparison with existing algorithms.

Contribution

It develops a perturbative framework up to the fourth order for covariance reconstruction from truncated data, enhancing analytical understanding and practical estimation.

Findings

Perturbative method effectively reconstructs covariance matrices.

Comparison shows competitive performance with existing iterative algorithms.

Analytic properties of perturbative terms are characterized.

Abstract

In this paper we propose a perturbative method for the reconstruction of the covariance matrix of a multinormal distribution, under the assumption that the only available information amounts to the covariance matrix of a spherically truncated counterpart of the same distribution. We expand the relevant equations up to the fourth perturbative order and discuss the analytic properties of the first few perturbative terms. We finally compare the proposed approach with an exact iterative algorithm (presented in Palombi et al. (2017)) in the hypothesis that the spherically truncated covariance matrix is estimated from samples of various sizes.