Sampling spiked Wishart eigenvalues

Efficient schemes for sampling from the eigenvalues of the Wishart distribution have recently been described for both the standard Wishart case (where the covariance matrix is the identity) and the spiked Wishart with a single spike (where the covariance matrix differs from the identity in a single entry on the diagonal). Here, we generalize these schemes to the spiked Wishart with an arbitrary number of spikes. This approach also applies to the spiked pseudo-Wishart distribution. We describe how to differentiate this procedure for the purposes of stochastic gradient descent, allowing the fitting of the eigenvalue distribution to some target distribution.