Wasserstein Covariance for Multiple Random Densities

A common feature of methods for analyzing samples of probability density functions is that they respect the geometry inherent to the space of densities. Once a metric is specified for this space, the Fr\'echet mean is typically used to quantify and visualize the average density from the sample. For one-dimensional densities, the Wasserstein metric is popular due to its theoretical appeal and interpretive value as an optimal transport metric, leading to the Wasserstein-Fr\'echet mean or barycenter as the mean density. We extend the existing methodology for samples of densities in two key directions. First, motivated by applications in neuroimaging, we consider dependent density data, where a $p$-vector of univariate random densities is observed for each sampling unit. Second, we introduce a Wasserstein covariance measure and propose intuitively appealing estimators for both fixed and diverging $p$, where the latter corresponds to continuously-indexed densities. We also give theory demonstrating consistency and asymptotic normality, while accounting for errors introduced in the unavoidable preparatory density estimation step. The utility of the Wasserstein covariance matrix is demonstrated through applications to functional connectivity in the brain using functional magnetic resonance imaging data and to the secular evolution of mortality for various countries.Comment: 12 pages, 4 figure

Graph theory applied to neuroimaging data reveals key functional connectivity alterations in brain of behavioral variant Frontotemporal Dementia subjects

Brain functional architecture and anatomical structure have been intensively studied to generate efficient models of its complex mechanisms. Functional alterations and cognitive impairments are the most investigated aspects in the recent clinical research as distinctive traits of neurodegeneration. Although specific behaviours are clearly associated to neurodegeneration, information flow breakdown within the brain functional network, responsible to deeply affect cognitive skills, remains not completely understood. Behavioural variant Frontotemporal Dementia (bvFTD) is the most common type of Frontotemporal degeneration, marked by behavioural disturbances, social instabilities and impairment of executive functions. Mathematical modelling offers effective tools to inspect deviations from physiological cognitive functions and connectivity alterations. As a popular recent methodology, graph theoretical approaches applied to imaging data expanded our knowledge of neurodegenerative disorders, although the need for unbiased metrics is still an open issue. In this thesis, we propose an integrated analysis of functional features among brain areas in bvFTD patients, to assess global connectivity and topological network alterations respect to the healthy condition, using a minimum spanning tree (MST) based-model to resting state functional MRI (rs-fMRI) data. Contrary to several graph theoretical approaches, dependent to arbitrary criteria (e.g., correlation thresholds, network density or a priori distribution), MST represents an unambiguous modelling solution, ensuring full reproducibility and robustness in different conditions. Our MSTs were obtained from wavelet correlation matrices derived from mean time series intensities, extracted from 116 regions of interest (ROIs) of 41 bvFTD patients and 39 healthy controls (HC), which underwent rs-fMRI. The resulting graphs were tested for global connectivity and topological differences between the two groups, by applying a Wilcoxon rank sum test with a significance level at 0.05 (nonparametric median difference estimates with 95% confidence interval). The same test was applied for methodological comparison between MST and other common graph theory methods. After methodological comparisons, our MST model achieved the best bvFTD/HC separation performances, without a priori assumptions. Direct MST comparison between bvFTD and healty controls revealed key brain functional architecture differences. Diseased subjects showed a linear-shape network configuration tendency, with high distance between nodes, low centrality parameter values, and a low exchange information capacity (i.e., low network integration) in MST parameters. Moreover, edge-level and node-level features (i.e., superhighways, and node degree and betweenness centrality) indicated a more complex scenario, showing some of the key bvFTD dysfunctions observed in large scale resting-state functional networks (default-mode (DMN), salience (SN), and executive (EN) networks), suggesting an underlying involvement of the limbic system in the observed functional deterioration. Functional isolation has been observed as a generalized process affecting the entire bvFTD network, showing brain macro-regions isolation, with homogeneous functional distribution of brain areas, longer distances between hubs, and longer within-lobe superhighways. Conversely, the HC network showed marked functional integration, where superhighways serve as shortcuts to connect areas from different brain macro-regions. The combination of this theoretical model with rs-fMRI data constitutes an effective method to generate a clear picture of the functional divergence between bvFTD and HCs, providing possible insights on the effects of frontotemporal neurodegeneration and compensatory mechanisms underlying characteristic bvFTD cognitive, social, and executive impairments

Méthodes mathématiques d’analyse d’image pour les études de population transversales et longitudinales

In medicine, large scale population analysis aim to obtain statistical information in order to understand better diseases, identify their risk factors, develop preventive and curative treatments and improve the quality of life of the patients.In this thesis, we first introduce the medical context of Alzheimer’s disease, recall some concepts of statistical learning and the challenges that typically occurwhen applied in medical imaging. The second part focus on cross-sectional studies,i.e. at a single time point. We present an efficient method to classify white matter lesions based on support vector machines. Then we discuss the use of manifoldlearning techniques for image and shape analysis. Finally, we present extensions ofLaplacian eigenmaps to improve the low-dimension representations of patients usingthe combination of imaging and clinical data. The third part focus on longitudinalstudies, i.e. between several time points. We quantify the hippocampus deformations of patients via the large deformation diffeomorphic metric mapping frameworkto build disease progression classifiers. We introduce novel strategies and spatialregularizations for the classification and identification of biomarkers.En médecine, les analyses de population à grande échelle ont pour but d’obtenir des informations statistiques pour mieux comprendre des maladies, identifier leurs facteurs de risque, développer des traitements préventifs et curatifs et améliorer la qualité de vie des patients.Dans cette thèse, nous présentons d’abord le contexte médical de la maladie d’Alzheimer, rappelons certains concepts d’apprentissage statistique et difficultés rencontrées lors de l’application en imagerie médicale. Dans la deuxième partie,nous nous intéressons aux analyses transversales, c-a-d ayant un seul point temporel.Nous présentons une méthode efficace basée sur les séparateurs à vaste marge (SVM)permettant de classifier des lésions dans la matière blanche. Ensuite, nous étudions les techniques d’apprentissage de variétés pour l’analyse de formes et d’images, et présentons deux extensions des Laplacian eigenmaps améliorant la représentation de patients en faible dimension grâce à la combinaison de données d’imagerie et cliniques. Dans la troisième partie, nous nous intéressons aux analyses longitudinales, c-a-d entre plusieurs points temporels. Nous quantifions les déformations des hippocampus de patients via le modèle des larges déformations par difféomorphismes pour classifier les évolutions de la maladie. Nous introduisons de nouvelles stratégies et des régularisations spatiales pour la classification et l’identification de marqueurs biologiques

Decentralized Algorithms for Wasserstein Barycenters

In dieser Arbeit beschäftigen wir uns mit dem Wasserstein Baryzentrumproblem diskreter Wahrscheinlichkeitsmaße sowie mit dem population Wasserstein Baryzentrumproblem gegeben von a Fréchet Mittelwerts von der rechnerischen und statistischen Seiten. Der statistische Fokus liegt auf der Schätzung der Stichprobengröße von Maßen zur Berechnung einer Annäherung des Fréchet Mittelwerts (Baryzentrum) der Wahrscheinlichkeitsmaße mit einer bestimmten Genauigkeit. Für empirische Risikominimierung (ERM) wird auch die Frage der Regularisierung untersucht zusammen mit dem Vorschlag einer neuen Regularisierung, die zu den besseren Komplexitätsgrenzen im Vergleich zur quadratischen Regularisierung beiträgt. Der Rechenfokus liegt auf der Entwicklung von dezentralen Algorithmen zurBerechnung von Wasserstein Baryzentrum: duale Algorithmen und Sattelpunktalgorithmen. Die Motivation für duale Optimierungsmethoden ist geschlossene Formen für die duale Formulierung von entropie-regulierten Wasserstein Distanz und ihren Derivaten, während, die primale Formulierung nur in einigen Fällen einen Ausdruck in geschlossener Form hat, z.B. für Gaußsches Maß. Außerdem kann das duale Orakel, das den Gradienten der dualen Darstellung für die entropie-regulierte Wasserstein Distanz zurückgibt, zu einem günstigeren Preis berechnet werden als das primale Orakel, das den Gradienten der (entropie-regulierten) Wasserstein Distanz zurückgibt. Die Anzahl der dualen Orakel rufe ist in diesem Fall ebenfalls weniger, nämlich die Quadratwurzel der Anzahl der primalen Orakelrufe. Im Gegensatz zum primalen Zielfunktion, hat das duale Zielfunktion Lipschitz-stetig Gradient aufgrund der starken Konvexität regulierter Wasserstein Distanz. Außerdem untersuchen wir die Sattelpunktformulierung des (nicht regulierten) Wasserstein Baryzentrum, die zum Bilinearsattelpunktproblem führt. Dieser Ansatz ermöglicht es uns auch, optimale Komplexitätsgrenzen zu erhalten, und kann einfach in einer dezentralen Weise präsentiert werden.In this thesis, we consider the Wasserstein barycenter problem of discrete probability measures as well as the population Wasserstein barycenter problem given by a Fréchet mean from computational and statistical sides. The statistical focus is estimating the sample size of measures needed to calculate an approximation of a Fréchet mean (barycenter) of probability distributions with a given precision. For empirical risk minimization approaches, the question of the regularization is also studied along with proposing a new regularization which contributes to the better complexity bounds in comparison with the quadratic regularization. The computational focus is developing decentralized algorithms for calculating Wasserstein barycenters: dual algorithms and saddle point algorithms. The motivation for dual approaches is closed-forms for the dual formulation of entropy-regularized Wasserstein distances and their derivatives, whereas the primal formulation has a closed-form expression only in some cases, e.g., for Gaussian measures.Moreover, the dual oracle returning the gradient of the dual representation forentropy-regularized Wasserstein distance can be computed for a cheaper price in comparison with the primal oracle returning the gradient of the (entropy-regularized) Wasserstein distance. The number of dual oracle calls in this case will be also less, i.e., the square root of the number of primal oracle calls. Furthermore, in contrast to the primal objective, the dual objective has Lipschitz continuous gradient due to the strong convexity of regularized Wasserstein distances. Moreover, we study saddle-point formulation of the non-regularized Wasserstein barycenter problem which leads to the bilinear saddle-point problem. This approach also allows us to get optimal complexity bounds and it can be easily presented in a decentralized setup

Remembering Forward: Neural Correlates of Memory and Prediction in Human Motor Adaptation

We used functional MR imaging (FMRI), a robotic manipulandum and systems identification techniques to examine neural correlates of predictive compensation for spring-like loads during goal-directed wrist movements in neurologically-intact humans. Although load changed unpredictably from one trial to the next, subjects nevertheless used sensorimotor memories from recent movements to predict and compensate upcoming loads. Prediction enabled subjects to adapt performance so that the task was accomplished with minimum effort. Population analyses of functional images revealed a distributed, bilateral network of cortical and subcortical activity supporting predictive load compensation during visual target capture. Cortical regions – including prefrontal, parietal and hippocampal cortices – exhibited trial-by-trial fluctuations in BOLD signal consistent with the storage and recall of sensorimotor memories or “states” important for spatial working memory. Bilateral activations in associative regions of the striatum demonstrated temporal correlation with the magnitude of kinematic performance error (a signal that could drive reward-optimizing reinforcement learning and the prospective scaling of previously learned motor programs). BOLD signal correlations with load prediction were observed in the cerebellar cortex and red nuclei (consistent with the idea that these structures generate adaptive fusimotor signals facilitating cancelation of expected proprioceptive feedback, as required for conditional feedback adjustments to ongoing motor commands and feedback error learning). Analysis of single subject images revealed that predictive activity was at least as likely to be observed in more than one of these neural systems as in just one. We conclude therefore that motor adaptation is mediated by predictive compensations supported by multiple, distributed, cortical and subcortical structures

Functional Magnetic Resonance Spectroscopy in First-Episode Schizophrenia: Measuring Glutamate and Glutathione Dynamics at 7-Tesla

Schizophrenia is a neuropsychiatric illness without known etiology or cure. Current efforts for symptom treatment still seem to leave a large portion of affected individuals without proper symptom management, with those experiencing symptom relief still having to wrestle with potential side-effects from medication trials. There has been growing evidence suggesting that glutamate and glutathione abnormalities hold major roles in development and manifestation of schizophrenia symptoms. Magnetic resonance spectroscopy (MRS) provides a non-invasive means to observe in-vivo brain chemistry, including glutamate and glutathione. By adding a functional component to an MRS paradigm (fMRS), such as the color-word Stroop task, it is possible to detect potential changes in brain metabolite levels in response to a cognitive stimulus. The objectives of this thesis were to determine whether a conventional short echo-time or longer echo-time semi-LASER sequence would be more suitable for measurements of an fMRS paradigm at 7.0-Tesla, and to apply this finding to the study of a group of never-medicated, first-episode schizophrenia (FES) individuals to track any potential abnormal glutamate and glutathione dynamics. Contrary to previous beliefs that the shortest achievable echo-time would produce the most measurement signal, results from this study found that a long echo time (TE=100ms) produced very similar quality of measurements, with further benefits of long echo time being the removal of any macromolecular signal contribution. Comparison of healthy controls (n = 25) to a FES population (n = 21) revealed no significant difference in resting or dynamic glutamate levels. However, resting and dynamic glutathione level were significantly different, suggesting potential glutathione regulation abnormalities, and, in extension, although not observed in this study, potential glutamate and glutamine abnormalities. Future fMRS studies should investigate glutamate and glutathione dynamics from longitudinal data of FES follow-ups as well as between specific sub-groups within FES

Blood and cerebrospinal fluid biomarkers for Alzheimer’s disease: from clinical to preclinical cohorts

Dementia is a major contributor to global morbidity, mortality and costs associated with health and social care. Alzheimer’s disease (AD) is a common pathology culminating in dementia, but it has a preclinical phase of one to two decades, with early brain deposition of amyloid and tau, followed by synaptic and neuronal degeneration. Early detection during the preclinical phase of AD might enable disease-modifying therapies to be applied during a window of opportunity in which they would be more likely to work. Currently the main biomarkers of AD pathology are neuroimaging markers, which can be costly, or cerebrospinal fluid markers, which require invasive sampling. Blood biomarkers would be relatively less invasive and could be a more cost-effective means for risk stratification, early detection, monitoring progression and measuring response to treatment. The work described here used sensitive assay technology including the Simoa digital immunoassay platform, in large and well-characterised cohorts, to examine candidate blood biomarkers linked to the core AD pathologies of amyloid, tau and neurodegeneration, as specified by the National Institute on Aging and Alzheimer’s Association 2018 research framework. Firstly, experiments on samples from a cognitive clinic cohort established the stability of the blood biomarkers Aβ40, Aβ42, total tau and neurofilament light chain (NFL – a marker of neurodegeneration) to multiple freeze-thaw cycles, and the optimal blood fraction to use for quantifying each of these biomarkers in onward studies. Secondly, an unique large preclinical cohort with life course data (Insight 46, the neuroscience sub-study of 502 individuals from the MRC National Survey of Health and Development; the 1946 British birth cohort) was used to examine the cross-sectional relationships between these blood biomarkers, neuroimaging biomarkers (18F-florbetapir amyloid PET, whole brain and hippocampal volumes, white matter hyperintensity volume and cortical thickness in an AD signature region) and cognitive performance (PACC: preclinical Alzheimer’s composite and its constituents). Through a collaboration with the University of Gothenburg, a novel liquid chromatography-mass spectrometry (LC-MS) method for quantification of plasma amyloid-β species was compared with the commercial Simoa assays in Insight 46. This was the first direct method comparison study of plasma amyloid-β species for the detection of preclinical cerebral amyloid deposition. It showed that the LC-MS method, when combined with age, sex and APOE #-4 carrier status, was able to distinguish PET amyloid status with an optimal (Youden’s cut point) sensitivity of 85.7% and specificity of 72.7%. The Simoa biomarkers of plasma total tau and serum NFL were confirmed to be potentially useful prognostic markers, as lower AD signature cortical thickness was associated with higher plasma total tau and serum NFL, lower whole brain volume was associated with higher plasma total tau, and higher ventricular volume was associated with higher serum NFL. Lower PACC scores were associated with higher serum NFL and lower scores for a paired associative memory test in particular were associated with higher plasma total tau and serum NFL. Thirdly, through a collaboration with Harvard University and the University of California San Diego, a new N-terminal tau biomarker was developed in CSF and plasma that showed good accuracy in distinguishing individuals with symptomatic CSF-defined AD pathology from healthy controls. Taken together, this work has demonstrated the impact of pre-analytical factors on measurements of AD blood biomarkers, validated these biomarkers as indicators of the core pathologies of AD and helped to develop a new tau blood biomarker in AD