Kernel density estimation on the rotation group and its application to crystallographic texture analysis

Abstract

We are concerned with kernel density estimation on the rotation group SO(3). We prove asymptotically optimal convergence rates for the minimax risk of the mean integrated squared error for different function classes including bandlimited functions, functions with bounded Sobolev norm and functions with polynomially decaying Fourier coefficients and give optimal kernel functions. Furthermore, we consider kernel density estimation with nonnegative kernel functions and prove analogous saturation behavior as it is known for the Euclidean case, i.e., the optimal minimax rate does not improve for smoothness classes of functions which are more than two times differentiable. We also benchmark several families of kernel functions with respect to their capability for kernel density estimation. To make our finding applicable, we give a fast algorithm for the computation of the kernel density estimator for large sampling sets and illustrate our theoretical findings by numerical experiments. Finally, we apply our results to answer a long standing question in crystallographic texture analysis on the number of orientation measurements needed to estimate the underlying orientation density function up to a given accuracy.