De-Individualizing fMRI Signals via Mahalanobis Whitening and Bures Geometry

Functional connectivity has been widely investigated to understand brain disease in clinical studies and imaging-based neuroscience, and analyzing changes in functional connectivity has proven to be valuable for understanding and computationally evaluating the effects on brain function caused by diseases or experimental stimuli. By using Mahalanobis data whitening prior to the use of dimensionality reduction algorithms, we are able to distill meaningful information from fMRI signals about subjects and the experimental stimuli used to prompt them. Furthermore, we offer an interpretation of Mahalanobis whitening as a two-stage de-individualization of data which is motivated by similarity as captured by the Bures distance, which is connected to quantum mechanics. These methods have potential to aid discoveries about the mechanisms that link brain function with cognition and behavior and may improve the accuracy and consistency of Alzheimer's diagnosis, especially in the preclinical stage of disease progression.

Approximation properties of slice-matching operators

Iterative slice-matching procedures are efficient schemes for transferring a source measure to a target measure, especially in high dimensions. These schemes have been successfully used in applications such as color transfer and shape retrieval, and are guaranteed to converge under regularity assumptions. In this paper, we explore approximation properties related to a single step of such iterative schemes by examining an associated slice-matching operator, depending on a source measure, a target measure, and slicing directions. In particular, we demonstrate an invariance property with respect to the source measure, an equivariance property with respect to the target measure, and Lipschitz continuity concerning the slicing directions. We furthermore establish error bounds corresponding to approximating the target measure by one step of the slice-matching scheme and characterize situations in which the slice-matching operator recovers the optimal transport map between two measures. We also investigate connections to affine registration problems with respect to (sliced) Wasserstein distances. These connections can be also be viewed as extensions to the invariance and equivariance properties of the slice-matching operator and illustrate the extent to which slice-matching schemes incorporate affine effects.

Measure transfer via stochastic slicing and matching

This paper studies iterative schemes for measure transfer and approximation problems, which are defined through a slicing-and-matching procedure. Similar to the sliced Wasserstein distance, these schemes benefit from the availability of closed-form solutions for the one-dimensional optimal transport problem and the associated computational advantages. While such schemes have already been successfully utilized in data science applications, not too many results on their convergence are available. The main contribution of this paper is an almost sure convergence proof for stochastic slicing-and-matching schemes. The proof builds on an interpretation as a stochastic gradient descent scheme on the Wasserstein space. Numerical examples on step-wise image morphing are demonstrated as well.