Sampling Spiked Wishart Eigenvalues

TL;DR

This paper extends sampling schemes for Wishart eigenvalues to multiple spikes, enabling more flexible modeling of covariance structures and differentiable procedures for optimization tasks.

Contribution

It generalizes existing sampling methods to handle multiple spikes in the Wishart distribution and introduces differentiation techniques for stochastic gradient descent.

Findings

Generalized sampling for multiple spikes

Applicable to pseudo-Wishart distribution

Enables differentiable eigenvalue sampling

Abstract

Efficient schemes for sampling from the eigenvalues of the Wishart distribution have recently been described for both the uncorrelated central case (where the covariance matrix is $\mathbf{I}$) and the spiked Wishart with a single spike (where the covariance matrix differs from $\mathbf{I}$ 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.