Riemannian Variance Filtering: An Independent Filtering Scheme for Statistical Tests on Manifold-Valued Data

Performing large scale hypothesis testing on brain imaging data to identify group-wise differences (e.g., between healthy and diseased subjects) typically leads to a large number of tests (one per voxel). Multiple testing adjustment (or correction) is necessary to control false positives, which may lead to lower detection power in detecting true positives. Motivated by the use of socalled 'independent filtering' techniques in statistics (for genomics applications), this paper investigates the use of independent filtering for manifold-valued data (e.g., Diffusion Tensor Imaging, Cauchy Deformation Tensors) which are broadly used in neuroimaging studies. Inspired by the concept of variance of a Riemannian Gaussian distribution, a type of non-specific data-dependent Riemannian variance filter is proposed. In practice, the filter will select a subset of the full set of voxels for performing the statistical test, leading to a more appropriate multiple testing correction. Our experiments on synthetic/simulated manifoldvalued data show that the detection power is improved when the statistical tests are performed on the voxel locations that 'pass' the filter. Given the broadening scope of applications where manifold-valued data are utilized, the scheme can serve as a general feature selection scheme.