Measure of distance between two probability distributions plays a fundamental role in statistics and machine learning. Optimal Transport (OT) theory provides such distance. Recent advance in OT theory is a generalization of classical OT with entropy regularized, called entropic OT. Despite its convenience in computation, it still lacks theoretical support. In this thesis, we study the connection between entropic OT and Entropy Power Inequality (EPI). First, we prove an HWI-type inequality making use of the infinitesimal displacement convexity of OT map. Second, we derive two Talagrand-type inequalities using the saturation of EPI that corresponds to a numerical term in our expression. We evaluate for a wide variety of distributions this term whereas for Gaussian and i.i.d. Cauchy distributions this term is found in explicit form. We show that our results extend previous results of Gaussian Talagrand inequality for Sinkhorn distance to the strongly log-concave case. Furthermore, we observe a dimensional measure concentration phenomenon using the new Talagrand-type inequality. / Mått på avstånd mellan två sannolikhetsfördelningar spelar en grundläggande roll i statistik och maskininlärning. Optimal transport (OT) teori ger ett sådant avstånd. Nyligen framsteg inom OT-teorin är en generalisering av klassisk OT med entropi-reglerad, kallad entropisk OT. Trots dess bekvämlighet i beräkning saknar det fortfarande teoretiskt stöd. I denna avhandling studerar vi sambandet mellan entropisk OT och Entropy Power Inequality (EPI). Först bevisar vi en ojämlikhet av HWI-typ med användning av OT-kartans oändliga förskjutningskonvexitet. För det andra härleder vi två Talagrand-typkvaliteter med mättnaden av EPI som motsvarar ett numeriskt uttryck vårt uttryck. Vi utvärderar för ett brett utbud av distributioner den här termen för Gauss och i.i.d. Cauchy-distributioner denna term finns oförklarlig form. Vi visar att våra resultat utökar tidigare resultat av GaussianTalagrand-ojämlikhet för Sinkhorn-avstånd till det starkt log-konkava fallet. Dessutom observerar vi ett dimensionellt mått koncentrationsfenomen mot den nya Talagrand-typen ojämlikhet.

Magnetic Resonance Imaging (MRI) is an essential healthcare technology, with diffusion MRI being a specialized technique. Diffusion MRI exploits the inherent diffusion of water molecules within the human body to produce a high-resolution tissue image. An MRI image contains information about a 3D volume in space, composed of 3D units called voxels. This thesis assumes the existence of a probability distribution for the diffusivity within a voxel, the diffusion tensor distribution (DTD), with the diffusivity described by a family of diffusion tensors. In 2D, these tensors can be described by 2x2 symmetric positive semidefinite matrices. The objective is to estimate the DTD of a voxel with neural networks for both 1D and 2D diffusion tensors. We assume the DTD to be a discrete distribution, with a finite set of diffusion tensors. The MRI signal is influenced by several experimental parameters, which for diffusion measurements are summarized in the measurement tensor B. To determine the diffusivity of a voxel, multiple measurement tensors are utilized, producing various MRI signals. From these signals, the network estimates the corresponding DTD of the voxel. The network seeks to employ the earth mover's distance (EMD) as its loss function, given the established use of EMD as a distance between probability distributions. Due to the difficulty of expressing the EMD as a differentiable loss function, the Sinkhorn distance, an entropic regularized approximation of the EMD, is used instead. Different network configurations are explored through simulations to identify optimal settings, assessed by the EMD loss and the closeness of the Sinkhorn loss to the EMD. The results indicate that the network achieves satisfactory accuracy when approximating DTDs with a small number of diffusivities, but struggles when the number increases. Future work could explore alternative loss functions and advanced neural network architectures. Despite the challenges encountered, this thesis offers relevant insight into the estimation of diffusion tensor distributions.