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In this paper we relate the matrix $S_B$ of the second moments of aspherically 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 bythe truncation, its eigenvalues are non-uniformly damped. We show that theeigenvalues of $\Sigma$ can be reconstructed from their truncated counterpartsvia a fixed point iteration, whose convergence we prove analytically. Theprocedure requires the computation of multidimensional Gaussian integrals overa Euclidean ball, for which we extend a numerical technique, originallyproposed by Ruben in 1962, based on a series expansion in chi-squaredistributions. In order to study the feasibility of our approach, we examinethe convergence rate of some iterative schemes on suitably chosen ensembles ofWishart matrices. We finally discuss the practical difficulties arising insample space and outline a regularization of the problem based on perturbationtheory.

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