Random Hyperplanes, Generalized Singular Values & “What’s My β?”

We streamline the treatment of the Jacobi ensemble from random matrix theory by providing a succinct geometric characterization which may be used directly to compute the Jacobi ensemble distribution without unnecessary matrix baggage traditionally seen in the MANOVA formulation. Algebraically the Jacobi ensemble naturally corresponds to the Generalized Singular Value Decomposition from the field of Numerical Linear Algebra. We further provide a clear geometric interpretation for the Selberg constant in front of the distribution which may sensibly be defined even beyond the reals, complexes, and quaternions. On the application side, we propose a new learning problem where one estimates a $\beta $ that best fits the sample eigenvalues from the Jacobi ensemble.