  This paper uses techniques from Random Matrix Theory to find the ideal
training-testing data split for a simple linear regression with m data points,
each an independent n-dimensional multivariate Gaussian. It defines "ideal" as
satisfying the integrity metric, i.e. the empirical model error is the actual
measurement noise, and thus fairly reflects the value or lack of same of the
model. This paper is the first to solve for the training and test size for any
model in a way that is truly optimal. The number of data points in the training
set is the root of a quartic polynomial Theorem 1 derives which depends only on
m and n; the covariance matrix of the multivariate Gaussian, the true model
parameters, and the true measurement noise drop out of the calculations. The
critical mathematical difficulties were realizing that the problems herein were
discussed in the context of the Jacobi Ensemble, a probability distribution
describing the eigenvalues of a known random matrix model, and evaluating a new
integral in the style of Selberg and Aomoto. Mathematical results are supported
with thorough computational evidence. This paper is a step towards automatic
choices of training/test set sizes in machine learning.
