Hicham will defend his PhD thesis on March 23rd at 10 am, online.
Title: Advances in Optimal transport and applications to Neuroscience
Brain imaging devices can provide a glimpse at neural activity in multiple spatial locations and time points. Moreover, neuroimaging studies are usually conducted for multiple individuals undergoing the same experimental protocol. Inferring the underlying sources is a challenging inverse problem that can only be tackled by biasing the solutions with prior domain knowledge. Several prior hypotheses have been pursued in the literature such as promoting sparse over dense solutions or solving the problem for multiple subjects at once. However, none take advantage of the particular spatial geometry of the problem. The purpose of this thesis is to exploit the multi-subject, spatial and temporal aspects of magneto-encephalography data as much as possible to improve the conditioning of the inverse problem. To that end, our contributions revolve around three axes: optimal transport (OT), sparse multi-task regression and time series. Indeed, the ability of OT to capture spatial disparities between measures makes it very well suited to compare and average neural activation patterns based on their shape and location over the cortical surface of the brain. For the sake of scalability, we take advantage of the entropic formulation of optimal transport, which we argue has two important missing pieces. From a theoretical perspective, it has no closed form analytical expressions, and from a practical perspective, entropy leads to a significant increase in variance known as entropic bias. We complete this puzzle by studying multivariate Gaussians for which we uncover an entropic OT closed form and propose debiased algorithms to compute fast and accurate optimal transport barycenters. Second, we define a multi-task prior based on OT and sparse penalties to jointly solve the inverse problem for multiple subjects to promote spatially coherent solutions. Our real data experiments highlight the benefits of using OT as a prior over classical multi-task regression penalties. Finally, we propose a loss function to compare and average spatio-temporal data that computes temporal alignments across spatially similar observations of the data via a fast GPU friendly algorithm.