Score matching estimators for directional distributions

TL;DR

This paper introduces a score matching estimation method for directional distributions on Riemannian manifolds, enabling efficient parameter estimation without computing complex normalizing constants.

Contribution

It develops a general score matching estimator applicable to directional models on manifolds, simplifying estimation and ensuring consistency and asymptotic normality.

Findings

Estimator is consistent and asymptotically normal.

Method avoids computing normalizing constants.

Demonstrated good performance through examples.

Abstract

One of the major problems for maximum likelihood estimation in the well-established directional models is that the normalising constants can be difficult to evaluate. A new general method of "score matching estimation" is presented here on a compact oriented Riemannian manifold. Important applications include von Mises-Fisher, Bingham and joint models on the sphere and related spaces. The estimator is consistent and asymptotically normally distributed under mild regularity conditions. Further, it is easy to compute as a solution of a linear set of equations and requires no knowledge of the normalizing constant. Several examples are given, both analytic and numerical, to demonstrate its good performance.