Sampling uniformly from the surface of an ellipsoid (in the sense of $\mu(dA) = \frac{1}{A}$) seems very nontrivial:

with most suggesting rejection sampling.

On the other hand, a common approach for sampling on the $d$-sphere uses the spherical symmetry of the standard multivariate Gaussian to use $\frac{v}{||v||}$. More generally, for a zero-mean multivariate Gaussian with covariance $\Sigma$, the ellipsoids $$ v^\intercal \Sigma^{-1}v = c^2 $$ are sets of constant density. Yet, the following naive approach does not work (I assume due to inhomogeneous change in the area element):

$v \sim \mathcal{N}(0, \Sigma)$, construct samples $x = \frac{v}{\sqrt{v^\intercal \Sigma^{-1}v}}$

Yet, can we not use the level sets of a multivariate gaussian somehow to sample an ellipsoid uniformly without rejection?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.