Toward a Data–Driven Test of Uniformity on an Unknown Riemannian Manifold

In this work we propose to extend a data–driven version of Gine' Sobolev test of uniformity due to Jupp to the case of an unknown compact Riemannian manifold M. The basic idea behind Gine's family of tests is to first map M into the Hilbert space L2(M,μ) of square–integrable real–valued functions on M by a function t:M → L2(M,μ) having zero expectation whenever the measure μ equals the uniform distribution on M , and then to reject uniformity if the sample mean of t(x) is “far” from 0. When M is completely specified, the standard way of constructing such mappings t(·) is based on the eigenfunctions of the Laplace–Beltrami operator of the manifold. In our setup the manifold itself is unknown, consequently we suggest to start the construction in the most natural way; that is, by replacing the L-B operator with the so called discrete graph Laplacian L. The convergence of L to the manifold Laplacian as the sample size diverges is the justification for the success of many recent algorithms in machine learning. In particular our approach is inspired by the so called Spectral Graph Wavelet Transform (SGWT) and here its finite sample performance are investigated through an extensive simulation study.