Avoiding high-precision ODE solving for vMF negative entropy in small dimensions

Determine practical computational methods, particularly for small dimensions D, to exploit the differential relations among the derivatives of the von Mises–Fisher negative entropy Psi(mu)—including the second-order nonlinear ODE psi''(r) = psi'(r) / ((1 - r^2) psi'(r) + (1 - D) r) for r = ||mu||—to accurately compute Psi(mu) and its gradient without requiring high-precision numerical integration of the ODE.

Background

The paper derives a second-order nonlinear ODE for the radial profile of the von Mises–Fisher (vMF) negative entropy Psi(mu) based on radial symmetry and a trace identity for the covariance on the unit hypersphere. This ODE provides a route to compute Psi(mu), its gradient, and the variance function numerically.

While the authors also present closed-form approximations that perform increasingly well with larger dimensions, they note that for small dimensions achieving accurate values can necessitate high-precision ODE solves. They explicitly identify as open how best to leverage the established derivative relations to avoid such high-precision numerical integration in the small-dimensional regime.