The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation

Comparing metric measure spaces (i.e. a metric space endowed with aprobability distribution) is at the heart of many machine learning problems. The most popular distance between such metric measure spaces is theGromov-Wasserstein (GW) distance, which is the solution of a quadratic assignment problem. The GW distance is however limited to the comparison of metric measure spaces endowed with a probability distribution.To alleviate this issue, we introduce two Unbalanced Gromov-Wasserstein formulations: a distance and a more tractable upper-bounding relaxation.They both allow the comparison of metric spaces equipped with arbitrary positive measures up to isometries. The first formulation is a positive and definite divergence based on a relaxation of the mass conservation constraint using a novel type of quadratically-homogeneous divergence. This divergence works hand in hand with the entropic regularization approach which is popular to solve large scale optimal transport problems. We show that the underlying non-convex optimization problem can be efficiently tackled using a highly parallelizable and GPU-friendly iterative scheme. The second formulation is a distance between mm-spaces up to isometries based on a conic lifting. Lastly, we provide numerical experiments onsynthetic examples and domain adaptation data with a Positive-Unlabeled learning task to highlight the salient features of the unbalanced divergence and its potential applications in ML

Effect of different spatial normalization approaches on tractography and structural brain networks

To facilitate the comparison of white matter morphologic connectivity across target populations, it is invaluable to map the data to a standardized neuroanatomical space. Here, we evaluated direct streamline normalization (DSN), where the warping was applied directly to the streamlines, with two publically available approaches that spatially normalize the diffusion data and then reconstruct the streamlines. Prior work has shown that streamlines generated after normalization from reoriented diffusion data do not reliably match the streamlines generated in native space. To test the impact of these different normalization methods on quantitative tractography measures, we compared the reproducibility of the resulting normalized connectivity matrices and network metrics with those originally obtained in native space. The two methods that reconstruct streamlines after normalization led to significant differences in network metrics with large to huge standardized effect sizes, reflecting a dramatic alteration of the same subject’s native connectivity. In contrast, after normalizing with DSN we found no significant difference in network metrics compared with native space with only very small-to-small standardized effect sizes. DSN readily outperformed the other methods at preserving native space connectivity and introduced novel opportunities to define connectome networks without relying on gray matter parcellations. Direct streamline normalization (DSN) directly warps the streamlines into any template space by using the transformations output from Advanced Normalization Tools (ANTs). DSN overcomes the limitations of diffusion weighted images (DWI) spatial normalization. It allows DWIs to be acquired with any desired sampling scheme. Fiber orientation distributions (FODs) or orientation distribution functions (ODFs) can also be reconstructed using any desired method and streamlines generated using any algorithm. Most importantly, it avoids the problem of generating tracts from FODs or ODFs that have become distorted because of spatial normalization. Our results show that DSN has minimal influence on tractography measures such as tract count and structure and does not significantly alter structural networks with only very small to small effect sizes. We have developed a framework in Python that works with most diffusion software platforms. It is available at http://github.com/clintg6/DSN

An automated pipeline for constructing personalized virtual brains from multimodal neuroimaging data

AbstractLarge amounts of multimodal neuroimaging data are acquired every year worldwide. In order to extract high-dimensional information for computational neuroscience applications standardized data fusion and efficient reduction into integrative data structures are required. Such self-consistent multimodal data sets can be used for computational brain modeling to constrain models with individual measurable features of the brain, such as done with The Virtual Brain (TVB). TVB is a simulation platform that uses empirical structural and functional data to build full brain models of individual humans. For convenient model construction, we developed a processing pipeline for structural, functional and diffusion-weighted magnetic resonance imaging (MRI) and optionally electroencephalography (EEG) data. The pipeline combines several state-of-the-art neuroinformatics tools to generate subject-specific cortical and subcortical parcellations, surface-tessellations, structural and functional connectomes, lead field matrices, electrical source activity estimates and region-wise aggregated blood oxygen level dependent (BOLD) functional MRI (fMRI) time-series. The output files of the pipeline can be directly uploaded to TVB to create and simulate individualized large-scale network models that incorporate intra- and intercortical interaction on the basis of cortical surface triangulations and white matter tractograpy. We detail the pitfalls of the individual processing streams and discuss ways of validation. With the pipeline we also introduce novel ways of estimating the transmission strengths of fiber tracts in whole-brain structural connectivity (SC) networks and compare the outcomes of different tractography or parcellation approaches. We tested the functionality of the pipeline on 50 multimodal data sets. In order to quantify the robustness of the connectome extraction part of the pipeline we computed several metrics that quantify its rescan reliability and compared them to other tractography approaches. Together with the pipeline we present several principles to guide future efforts to standardize brain model construction. The code of the pipeline and the fully processed data sets are made available to the public via The Virtual Brain website (thevirtualbrain.org) and via github (https://github.com/BrainModes/TVB-empirical-data-pipeline). Furthermore, the pipeline can be directly used with High Performance Computing (HPC) resources on the Neuroscience Gateway Portal (http://www.nsgportal.org) through a convenient web-interface

Analyse de l'architecture de la matière blanche et projection de mesures sur la surface corticale

L'étude de l'architecture et de la connectivité structurelle du cerveau est possible grâce à l'imagerie par résonance magnétique de diffusion (IRMd). Ce type d’image, similaire à un champ vectoriel tridimensionnel, combiné à un algorithme nommé tractographie, permet d’inférer la distribution des fibres de matière blanche et ainsi de reconstruire la structure locale du tissu. Or, cette méthode demeure limitée par une basse résolution et un faible rapport signal sur bruit. Afin de contourner ces limitations, des modèles géométriques construits à partir d’aprioris anatomiques sont utilisés. Cette thèse montre que des règles et des contraintes basées sur la modélisation corticale peuvent être intégrées à la tractographie par le biais d’équations de géométrie différentielle. En effet, la structure axonale sous-jacente à la matière grise peut être approximée avec l'utilisation de la surface et d'un flot de courbure moyenne. Pondéré par l’information de densité, ce flot permet d’obtenir une meilleure représentation des projections des fibres de matière blanche sous le cortex. D'ailleurs, le fait d’incorporer la surface corticale, obtenue d’une image anatomique haute résolution, à l’IRMd permet d'augmenter la précision de la tractographie. Puisque l'acquisition d'une image anatomique (pondération T1) est toujours faite lors d'une IRMd, la combinaison des deux est une façon simple et peu coûteuse d'améliorer cette technique de reconstruction. Par ailleurs, discrétiser les surfaces corticales à l'aide de maillages, plutôt qu’avec des masques voxeliques, permet non seulement d'augmenter la précision de l'interface, mais d'intégrer facilement de nouveaux aprioris et de mieux choisir la répartition des positions initiales. L'ajout d'aprioris et de modèles géométriques permet de mieux modéliser près du cortex et ainsi connecter jusqu'aux surfaces corticales. Cette connexion rend possible la projection de mesures de la matière blanche le long du cortex, un domaine également utilisé pour plusieurs analyses anatomiques (ex. épaisseur corticale), magnéto-/électro-encéphalographie (MEG/EEG) et IRM fonctionnelle (IRMf). L'intégration de ces surfaces corticales à la tractographie a un impact important pour les recherches multimodales sur la connectivité cérébrale