Numerical reconstruction of the covariance matrix of a spherically truncated multinormal distribution

TL;DR

This paper introduces a method to reconstruct the full covariance matrix of a multivariate normal distribution from its truncated version, using eigenvalue correction and numerical integration techniques.

Contribution

It presents a novel algorithm to invert the relation between truncated and full covariance matrices, including eigenvalue reconstruction via fixed point iteration and numerical integration extension.

Findings

Eigenvectors remain invariant under truncation.

Eigenvalues can be reconstructed through fixed point iteration.

The method's convergence is analytically proven.

Abstract

In this paper we relate the matrix $S_B$ of the second moments of a spherically truncated normal multivariate to its full covariance matrix $\Sigma$ and present an algorithm to invert the relation and reconstruct $\Sigma$ from $S_B$. While the eigenvectors of $\Sigma$ are left invariant by the truncation, its eigenvalues are non-uniformly damped. We show that the eigenvalues of $\Sigma$ can be reconstructed from their truncated counterparts via a fixed point iteration, whose convergence we prove analytically. The procedure requires the computation of multidimensional Gaussian integrals over a Euclidean ball, for which we extend a numerical technique, originally proposed by Ruben in 1962, based on a series expansion in chi-square distributions. In order to study the feasibility of our approach, we examine the convergence rate of some iterative schemes on suitably chosen ensembles of Wishart matrices. We finally discuss the practical difficulties arising in sample space and outline a regularization of the problem based on perturbation theory.