Although relativistic physics tend to omit nondynamical "absolute objects" such as a flat metric tensor or a preferred time foliation, there exist interesting questions related to such entities, such as worries about the "flow" of time in special relativity, and the apparent disappearance of time altogether in canonical general relativity. This latter problem is related to the lack of a fixed causal structure with repect to which one might posit "equal-time" commutation relations, for example. In view of these issues, we consider whether including a flat background metric, and perhaps a preferred foliation, is physically worthwhile. We show how a derivation of Einstein's equations from flat spacetime can be generalized to include a preferred foliation, the possible significance of which we discuss, though ultimately we suggest why such a foliation might be present in metaphysics and yet absent from physics. We also derive a new "slightly bimetric" class of theories using the flat spacetime approach. However, such derivations are only formally special relativistic, because they give no heed to the flat metric's causal structure, which the curved effective metric might well violate. After reviewing the history of this problem, we introduce new variables to give a kinematic description of the relation between the two null cones. Then we propose a method to enforce special relativistic causality by using the guage freedom to restrict the configuration space suitably. Consequences for exact solutions, such as the Schwarzschild solution and its 'singularity,' are discussed. Advantages and difficulties regarding adding a mass term to the theory are discussed briefly. / text

This thesis details two experimental tests that can be applied to particular quantum states to reveal important information. We begin by discussing the relevant background in quantum information. We introduce qubits and qudits as basic quantum states, and we discuss the evolution and measurement of quantum states. We then discuss quantum state tomography as a means by which to obtain complete information about a state, followed by a discussion of state discrimination as a means by which to determine the state given the promise that it is drawn from some known set. We then discuss relevant experimental techniques in quantum optics, including measurement, generation of entanglement, and generation of single photons from entanglement. The first experiment we discuss deals with the causal structure of a system, which is the description of the origin of correlations between two or more states. The causal structure can be direct-cause, meaning that one state causes the other; common-cause, meaning that both states are caused by another; or hybrid-cause, which is a combination of the two. We perform the first implementation of a new type of tomography to determine the causal structure; this is called causal tomography and functions regardless of whether two qubits are related by a common state, a process, or some combination thereof. We implement a process on two entangled photons so that we can select the exact causal structure that results, which ranges continuously between direct-cause and common-cause structures. Using causal tomography, we recover causal structures that closely match expected results and demonstrate that quantum mechanics provides an advantage in causal inference. The second experiment we discuss deals with the unambiguous discrimination of multiple quantum states. For the first time, we apply the principles of unambiguous state discrimination to high-dimensional systems. Given a state chosen randomly out of d possible states encoded in d dimensions, we implement a procedure for determining which state was chosen; this procedure in theory functions without error. We encode and detect the states in the orbital angular momentum degree of freedom up to dimension d=14. Although no experiment can provide perfectly error-free measurement due to inevitable imperfections, we obtain an error rate below the theoretical error rate of minimum-error state discrimination for dimensions up to d=12. At the time of submission of this thesis, this work has been accepted for publication in Physical Review Letters.

<div>During the past decades, deep learning has been proven to be an important tool for statistical data analysis. Motivated by the promise of deep learning in tackling the curse of dimensionality, we propose three innovative methods which apply deep learning techniques to high-dimensional data analysis in this dissertation.</div><div><br></div><div>Firstly, we propose a nonlinear sufficient dimension reduction method, the so-called split-and-merge deep neural networks (SM-DNN), which employs the split-and-merge technique on deep neural networks to obtain nonlinear sufficient dimension reduction of the input data and then learn a deep neural network on the dimension reduced data. We show that the DNN-based dimension reduction is sufficient for data drawn from exponential family, which retains all information on response contained in the explanatory data. Our numerical experiments indicate that the SM-DNN method can lead to significant improvement in phenotype prediction for a variety of real data examples. In particular, with only rare variants, we achieved a remarkable prediction accuracy of over 74\% for the Early-Onset Myocardial Infarction (EOMI) exome sequence data. </div><div><br></div><div>Secondly, we propose another nonlinear SDR method based on a new type of stochastic neural network under a rigorous probabilistic framework and show that it can be used for sufficient dimension reduction for high-dimensional data. The proposed stochastic neural network can be trained using an adaptive stochastic gradient Markov chain Monte Carlo algorithm. Through extensive experiments on real-world classification and regression problems, we show that the proposed method compares favorably with the existing state-of-the-art sufficient dimension reduction methods and is computationally more efficient for large-scale data.</div><div><br></div><div>Finally, we propose a structure learning method for learning the causal structure hidden in the high-dimensional data, which consists of two stages:</div><div>we first conduct Bayesian sparse learning for variable screening to build a primary graph, and then we perform conditional independence tests to refine the primary graph. </div><div>Extensive numerical experiments and quantitative tests confirm the generality, effectiveness and power of the proposed methods.</div>

This thesis explores the application of Graph Neural Networks (GNNs), a subset of Geometric Deep Learning methods, for medical image analysis and causal structure learning. Tracking the progression of pathologies in chest radiography poses several challenges in anatomical motion estimation and image registration as this task requires spatially aligning the sequential X-rays and modelling temporal dynamics in change detection. The first part of this thesis proposes a novel approach for change detection in sequential Chest X-ray (CXR) scans using GNNs. The proposed model CheXRelNet utilizes local and global information in CXRs by incorporating intra-image and inter-image anatomical information and showcases an increased downstream performance for predicting the change direction for a pair of CXRs. The second part of the thesis focuses on using GNNs for causal structure learning. The proposed method introduces the concept of intervention on graphs and attempts to relate belief propagation in Bayesian Networks (BN) to message passing in GNNs. Specifically, the proposed method leverages the downstream prediction accuracy of a GNN-based model to infer the correctness of Directed Acyclic Graph (DAG) structures given observational data. Our experimental results do not reveal any correlation between the downstream prediction accuracy of GNNs and structural correctness and hence indicate the harms of directly relating message passing in GNNs to belief propagation in BNs. Overall, this thesis demonstrates the potential of GNNs in medical image analysis and highlights the challenges and limitations of applying GNNs to causal structure learning. / Master of Science / Graphs are a powerful way to represent different real-world data such as interactions between patient observations, co-morbidities, treatments, and relationships between different parts of the human anatomy. They are also a simple and intuitive way of representing causeand- effect relationships between related entities. Graph Neural Networks (GNNs) are neural networks that model such graph-structured data. In this thesis, we explore the applicability of GNNs in analyzing chest radiography and in learning causal relationships. In the first part of this thesis, we propose a method for monitoring disease progression over time in sequential chest X-rays (CXRs). This proposed model CheXRelNet focuses on the interactions within different regions of a CXR and temporal interactions between the same region compared in two CXRs taken at different times for a given patient and accurately predicts the disease progression trend. In the second part of the thesis, we explore if GNNs can be used for identifying causal relationships between covariates. We design a method that uses GNNs for ranking different graph structures based on how well the structures explain the observed data.

Categorical quantum mechanics is a way of formalising the structural features of quantum theory using category theory. It uses compound systems as the primitive notion, which is formalised by using symmetric monoidal categories. This leads to an elegant formalism for describing quantum protocols such as quantum teleportation. In particular, categorical quantum mechanics provides a graphical calculus that exposes the information flow of such protocols in an intuitive way. However, the graphical calculus also reveals surprising features of these protocols; for example, in the quantum teleportation protocol, information appears to flow `backwards-in-time'. This leads to question of how causal structure can be described within categorical quantum mechanics, and how this might lead to insight regarding the structural compatibility between quantum theory and relativity. This thesis is concerned with the project of formalising causal structure in categorical quantum mechanics. We begin by studying an abstract view of Bell-type experiments, as described by `no-signalling boxes', and we show that under time-reversal no-signalling boxes generically become signalling. This conflicts with the underlying symmetry of relativistic causal structure. This leads us to consider the framework of categorical quantum mechanics from the perspective of relativistic causal structure. We derive the properties that a symmetric monoidal category must satisfy in order to describe systems in such a background causal structure. We use these properties to define a new type of category, and this provides a formal framework for describing protocols in spacetime. We explore this new structure, showing how it leads to an understanding of the counter-intuitive information flow of protocols in categorical quantum mechanics. We then find that the formal properties of our new structure are naturally related to axioms for reconstructing quantum theory, and we show how a reconstruction scheme based on purification can be formalised using the structures of categorical quantum mechanics. Finally, we discuss the philosophical aspects of using category theory to describe fundamental physics. We consider a recent argument that category-theoretic formulations of physics, such as categorical quantum mechanics, can be used to support a variant of structural realism. We argue against this claim. The work of this thesis suggests instead that the philosophy of categorical quantum mechanics is subtler than either operationalism or realism.

The pronounced constraints on ecosystem functioning and human livelihoods in drylands are frequently exacerbated by natural and socio-economic stresses, including weather extremes and inequitable trade conditions. Therefore, a better understanding of the relation between these stresses and the socio-ecological systems is important for advancing dryland development. The concept of vulnerability as applied in this dissertation describes this relation as encompassing the exposure to climate, market and other stresses as well as the sensitivity of the systems to these stresses and their capacity to adapt. With regard to the interest in improving environmental and living conditions in drylands, this dissertation aims at a meaningful generalisation of heterogeneous vulnerability situations. A pattern recognition approach based on clustering revealed typical vulnerability-creating mechanisms at global and local scales. One study presents the first analysis of dryland vulnerability with global coverage at a sub-national resolution. The cluster analysis resulted in seven typical patterns of vulnerability according to quantitative indication of poverty, water stress, soil degradation, natural agro-constraints and isolation. Independent case studies served to validate the identified patterns and to prove the transferability of vulnerability-reducing approaches. Due to their worldwide coverage, the global results allow the evaluation of a specific system’s vulnerability in its wider context, even in poorly-documented areas. Moreover, climate vulnerability of smallholders was investigated with regard to their food security in the Peruvian Altiplano. Four typical groups of households were identified in this local dryland context using indicators for harvest failure risk, agricultural resources, education and non-agricultural income. An elaborate validation relying on independently acquired information demonstrated the clear correlation between weather-related damages and the identified clusters. It also showed that household-specific causes of vulnerability were consistent with the mechanisms implied by the corresponding patterns. The synthesis of the local study provides valuable insights into the tailoring of interventions that reflect the heterogeneity within the social group of smallholders. The conditions necessary to identify typical vulnerability patterns were summarised in five methodological steps. They aim to motivate and to facilitate the application of the selected pattern recognition approach in future vulnerability analyses. The five steps outline the elicitation of relevant cause-effect hypotheses and the quantitative indication of mechanisms as well as an evaluation of robustness, a validation and a ranking of the identified patterns. The precise definition of the hypotheses is essential to appropriately quantify the basic processes as well as to consistently interpret, validate and rank the clusters. In particular, the five steps reflect scale-dependent opportunities, such as the outcome-oriented aspect of validation in the local study. Furthermore, the clusters identified in Northeast Brazil were assessed in the light of important endogenous processes in the smallholder systems which dominate this region. In order to capture these processes, a qualitative dynamic model was developed using generalised rules of labour allocation, yield extraction, budget constitution and the dynamics of natural and technological resources. The model resulted in a cyclic trajectory encompassing four states with differing degree of criticality. The joint assessment revealed aggravating conditions in major parts of the study region due to the overuse of natural resources and the potential for impoverishment. The changes in vulnerability-creating mechanisms identified in Northeast Brazil are well-suited to informing local adjustments to large-scale intervention programmes, such as “Avança Brasil”. Overall, the categorisation of a limited number of typical patterns and dynamics presents an efficient approach to improving our understanding of dryland vulnerability. Appropriate decision-making for sustainable dryland development through vulnerability reduction can be significantly enhanced by pattern-specific entry points combined with insights into changing hotspots of vulnerability and the transferability of successful adaptation strategies. / Die Grenzen ökologischer Funktionen und menschlicher Lebensweisen in Trockengebieten werden häufig durch natürlichen und sozio-ökonomischen Stress, wie extreme Wetterereignisse und ungerechte Handelsbedingungen, weiter verengt. Zur Förderung der Entwicklung in Trockengebieten ist es daher wichtig, die Beziehung zwischen den Stressfaktoren und den sozio-ökologischen Systemen besser zu verstehen. Das Konzept der Vulnerabilität, welches in der vorliegenden Dissertation angewandt wird, beschreibt dieses Verhältnis durch die Exposition, Sensitivität und Anpassungsfähigkeit von Systemen im Hinblick auf Klima-, Markt- und anderen Stress. Bezüglich des Interesses, die Umwelt- und Lebensbedingungen in Trockengebieten zu verbessern, zielt diese Dissertation darauf ab, die vielschichtigen Ursachen und Veränderungen von Vulnerabilität sinnvoll zu verallgemeinern. Eine clusterbasierte Mustererkennung zeigte typische Mechanismen auf, welche Vulnerabilität auf globaler und lokaler Ebene verursachen. Dabei stellt die globale Studie die erste flächendeckende Untersuchung von Vulnerabilität in Trockengebieten mit sub-nationaler Auflösung dar. Die Clusteranalyse identifizierte sieben typische Muster basierend auf der quantitativen Beschreibung von Armut, Wasserknappheit, Bodendegradation, natürlichen Produktionshemmnissen und Isolation. Die Gültigkeit der ermittelten Cluster und die Übertragbarkeit von Anpassungsmaßnahmen innerhalb ähnlicher Gebiete wurden anhand unabhängiger Fallstudien belegt. Die flächendeckende Erfassung erlaubt es, die Vulnerabilität eines Systems in seinem größeren Kontext zu bewerten, auch in weniger gut durch Fallstudien dokumentierten Gebieten. Weiterhin wurde die Klimavulnerabilität von Kleinbauern bezüglich ihrer Nahrungsmittelsicherung im peruanischen Altiplano untersucht. In diesem lokalen Kontext wurden vier Cluster von Haushalten gemäß ihrer Produktionsrisiken, landwirtschaftlichen Ressourcen, der Bildung und ihres nicht-landwirtschaftlichen Einkommens unterschieden. Eine erweiterte Gültigkeitsprüfung unter Nutzung unabhängig erhobener Informationen stellte heraus, dass wetterbedingte Schäden mit den ermittelten Clustern korrelieren und dass haushaltsspezifische Schadensursachen mit den durch die Muster angezeigten Mechanismen übereinstimmen. Die lokale Studie liefert wertvolle Hinweise auf bedarfsgerechte Eingriffe unter Beachtung der Heterogenität innerhalb der sozialen Gruppe der Kleinbauern. Die notwendigen Bedingungen zur Erkennung typischer Muster ergaben fünf methodische Schritte. Ihre Darlegung soll die Anwendung der gewählten Methode in zukünftigen Vulnerabilitätsstudien anregen und erleichtern. Die fünf Schritte umfassen die Ableitung relevanter Ursache-Wirkungs-Hypothesen, die Quantifizierung der Mechanismen, die Bewertung von Robustheit und Gültigkeit sowie die Ordnung der ermittelten Muster nach dem Grad der Vulnerabilität. Dabei ist die genaue Beschreibung der Hypothesen eine wesentliche Voraussetzung für die Quantifizierung der grundlegenden Prozesse sowie eine einheitliche Interpretation, Gültigkeitsprüfung und Ordnung der ermittelten Muster. Besondere Beachtung finden skalenbedingte Aspekte, wie beispielsweise die ergebnisorientierte Gültigkeitsprüfung in der lokalen Studie. Weiterhin wurden die in Nordostbrasilien ermittelten Cluster im Hinblick auf wichtige endogene Prozesse in den dort vorherrschenden kleinbäuerlichen Nutzungssystemen untersucht. Diese Prozesse umfassen die Aufteilung der Arbeitskraft, die landwirtschaftliche Produktion sowie Einkommens- und Ressourcendynamiken. Sie wurden in einem qualitativen dynamischen Modell erfasst, welches eine zyklische Trajektorie mit vier unterschiedlich problematischen Entwicklungszuständen ergab. Als besonders problematischer Aspekt verschärfte sich die Vulnerabilität in weiten Teilen des Untersuchungsgebietes durch die Übernutzung natürlicher Ressourcen und die Möglichkeit weiterer Verarmung. Die in Nordostbrasilien gezeigten Veränderungen sind dazu geeignet, groß angelegte Entwicklungsprogramme, wie zum Beispiel “Avança Brasil”, angemessen an lokale Gegebenheiten anzupassen. Insgesamt ermöglicht es die Kategorisierung einer begrenzten Anzahl typischer Muster und Veränderungen, die Vulnerabilität in Trockengebieten besser zu verstehen. Eine nachhaltige Entwicklung von Trockengebieten basierend auf der Minderung von Vulnerabilität kann durch musterspezifische Ansätze zusammen mit Hinweisen zu Veränderungen im Schweregrad und zur Übertragbarkeit erfolgreicher Anpassungsstrategien wirkungsvoll unterstützt werden.

This work is focused on visualisation of Schwarzschild, Reissner- Nordström and Kerr black hole. The two-dimensional conformal diagram was constructed. In the case of Kerr black hole, the causal structure was visualized by intersection of chronological future of given point in spacetime with hyper- surfaces of constant value of Boyer-Lindquist coordinate t. Conformal diagram for Kerr black hole was constructed only in the neighbourhood of outer event horizon. Then the causal diagram, which is analogous to conformal diagram for Reissner-Nordström black hole was constructed. In all cases two-dimensional spa- celike hypersurfaces were chosen that were embedded into Euclidean space. The interpretation of time evolution of black hole universe was given to a sequence of such embedded hypersurfaces. In the case of Kerr black hole the embedding of outer ergosphere and outer event horizon were also constructed. 1

Dans ce mémoire par articles, nous nous intéressons à l’apprentissage de modèles causaux à partir de données. L’intérêt de cette entreprise est d’obtenir une meilleure compréhension des données et de pouvoir prédire l’effet qu’aura un changement sur certaines variables d’un système étudié. Comme la découverte de liens causaux est fondamentale en sciences, les méthodes permettant l’apprentissage de modèles causaux peuvent avoir des applications dans une pléthore de domaines scientifiques, dont la génomique, la biologie et l’économie. Nous présentons deux nouvelles méthodes qui ont la particularité d’être des méthodes non-linéaires d’apprentissage de modèles causaux qui sont posées sous forme d’un problème d’optimisation continue sous contrainte. Auparavant, les méthodes d’apprentissage de mo- dèles causaux abordaient le problème de recherche de graphes en utilisant des stratégies de recherche voraces. Récemment, l’introduction d’une contrainte d’acyclicité a permis d’abor- der le problème différemment. Dans un premier article, nous présentons une de ces méthodes: GraN-DAG. Sous cer- taines hypothèses, GraN-DAG permet d’apprendre des graphes causaux à partir de données observationnelles. Depuis la publication du premier article, plusieurs méthodes alternatives ont été proposées par la communauté pour apprendre des graphes causaux en posant aussi le problème sous forme d’optimisation continue avec contrainte. Cependant, aucune de ces méthodes ne supportent les données interventionnelles. Pourtant, les interventions réduisent le problème d’identifiabilité et permettent donc l’utilisation d’architectures neuronales plus expressives. Dans le second article, nous présentons une autre méthode, DCDI, qui a la particularité de pouvoir utiliser des données avec différents types d’interventions. Comme le problème d’identifiabilité est moins important, une des deux instanciations de DCDI est un approximateur de densité universel. Pour les deux méthodes proposées, nous montrons que ces méthodes ont de très bonnes performances sur des données synthétiques et réelles comparativement aux méthodes traditionelles. / In this thesis by articles, we study the learning of causal models from data. The goal of this entreprise is to gain a better understanding of data and to be able to predict the effect of a change on some variables of a given system. Since discovering causal relationships is fundamental in science, causal structure learning methods have applications in many fields that range from genomics, biology, and economy. We present two new methods that have the particularity of being non-linear methods learning causal models casted as a continuous optimization problem subject to a constraint. Previously, causal strutural methods addressed this search problem by using greedy search heuristics. Recently, a new continuous acyclity constraint has allowed to address the problem differently. In the first article, we present one of these non-linear method: GraN-DAG. Under some assumptions, GraN-DAG can learn a causal graph from observational data. Since the publi- cation of this first article, several alternatives methods have been proposed by the community by using the same continuous-constrained optimization formulation. However, none of these methods support interventional data. Nevertheless, interventions reduce the identifiability problem and allow the use of more expressive neural architectures. In the second article, we present another method, DCDI, that has the particularity to leverage data with several kinds of interventions. Since the identifiabiliy issue is less severe, one of the two instantia- tions of DCDI is a universal density approximator. For both methods, we show that these methods have really good performances on synthetic and real-world tasks comparatively to other classical methods.

Linear time-invariant (LTI) dynamic networks are described by their dynamical structure function, and generally, they have many possible state space realizations. This work characterizes the necessary and sufficient conditions on a state transformation that preserves the dynamical structure function, thereby generating the entire set of realizations of a given order for a specific dynamic network.

Linear time-invariant (LTI) dynamic networks are described by their dynamical structure function, and generally, they have many possible state space realizations. This work characterizes the necessary and sufficient conditions on a state transformation that preserves the dynamical structure function, thereby generating the entire set of realizations of a given order for a specific dynamic network.