The Probability Density Function of a Transformation-based Hyperellipsoid Sampling Technique

TL;DR

This paper provides a rigorous proof that transforming uniform samples from an n-ball yields a uniform distribution over the hyperellipsoid, clarifying a previously unproven assumption in sampling techniques.

Contribution

It offers a formal proof confirming that the transformation-based sampling method produces a uniform distribution over hyperellipsoids from n-ball samples.

Findings

Confirmed the uniformity of the transformed samples

Validated the transformation technique mathematically

Clarified assumptions in hyperellipsoid sampling methods

Abstract

Sun and Farooq [2] showed that random samples can be efficiently drawn from an arbitrary n-dimensional hyperellipsoid by transforming samples drawn randomly from the unit n-ball. They stated that it was a straightforward to show that, given a uniform distribution over the n-ball, the transformation results in a uniform distribution over the hyperellipsoid, but did not present a full proof. This technical note presents such a proof.