A map estimation algorithm for Bayesian polynomial regression on riemannian manifolds

In this paper, we present a Bayesian formulation of polynomial regression on a Riemannian manifold. Previous methods for fitting a curve to manifold-valued data have been formulated as geometric, least-squares estimation problems. We show that least-squares estimation on manifolds, much like the familiar Euclidean case, suffers from overfitting when using higher-order polynomials. Our Bayesian model mitigates this overfitting by placing a prior on the polynomial coefficients that shrinks their magnitude, analogous to Bayesian Euclidean regression with a Gaussian prior on the coefficients. We develop an algorithm for computing maximum a posteriori estimates of polynomial coefficients and the noise variance. Experiments on synthetically generated sphere data and a real shape regression problem demonstrate the advantages of our approach.