Robust estimation of statistical parameters is traditionally believed to exist in a trade space between robustness and efficiency. This thesis examines the Minimum Hellinger Distance Estimator (MHDE), which is known to have desirable robustness properties as well as desirable efficiency properties. This thesis confirms that the MHDE is simultaneously robust against outliers and asymptotically efficient in the univariate location case. Robustness results are then extended to the case of simple linear regression, where the MHDE is shown empirically to have a breakdown point of 50%. A geometric algorithm for solution of the MHDE is developed and implemented. The algorithm utilizes the Riemannian manifold properties of the statistical model to achieve an algorithmic speedup. The MHDE is then applied to an illustrative problem in power system state estimation. The power system is modeled as a structured linear regression problem via a linearized direct current model; robustness results in this context have been investigated and future research areas have been identified from both a statistical perspective as well as an algorithm design standpoint. / Master of Science

As últimas décadas testemunharam intensa atividade de pesquisa teórica e experimental visando compreender o conceito do tempo na mecânica quântica. Este tema desencadeou significante progresso na busca por dispositivos mais rápidos e eficientes no processamento de informação e implementação de tecnologias de comunicação. Motivados pela pergunta quão rápido um sistema quântico evolui sob uma dada dinâmica?, tais avanços levaram a formulação do chamado limite quântico de velocidade ou quantum speed limit, (QSL), i.e., um limite inferior definindo o tempo mínimo de evolução entre estados quânticos distintos. Diversos resultados reportaram QSLs obtidos via tratamentos diferentes e aparentemente desconexos, muitas vezes sob configurações específicas, que deixaram uma lacuna fundamental à resposta da questão geral colocada anteriormente. Neste projeto investigamos como a não-unicidade de uma medida de distinguibilidade de operadores densidade definida no espaço de estados quânticos influencia o QSL e pode ser explorada no intuito de obter limites inferiores mais robustos no tempo de evolução de estados arbitrários. Em particular, baseando-nos no formalismo da geometria da informação, estabelecemos uma família infinita de QSLs válidos para evoluções unitárias e não-unitárias. Este trabalho se propõe unificar e generalizar resultados existentes sobre QSLs na literatura, além de fornecer exemplos de limites mais precisos do que aqueles baseados na informação de Fisher convencional. Em termos físicos, esta investigação é a primeira a destacar o papel das populações e coerências quânticas no cálculo e saturação dos QSLs. Nossos resultados podem encontrar aplicações na otimização de protocolos em computação quântica e metrologia, além de fornecer novos pontos de vista em investigações fundamentais da termodinâmica quântica. / The last decades witnessed intense theoretical and experimental research activity in order to understand the concept of time in quantum mechanics. This subject triggered significant progress in the search for faster and efficient schemes in the implementation of quantum information and communication technologies. Starting from the puzzle How fast can a quantum state evolve under a given dynamics?, such advances have led to the establishment of quantum speed limits (QSLs), i.e., a lower bound setting the minimum time evolution between two distinct quantum states. Past results have included different, apparently unrelated approaches to quantum speed limits, and sometimes tailored to specific settings, which therefore left a fundamental gap in obtaining a satisfactory answer to the general question posed above. In this work we provide a breakthrough for the study and applications of quantum speed limits. We approach the problem from a general information theoretic point of view and we adopt an elegant geometric formalism to construct an infinite family of quantum speed limits valid for closed and open system evolutions. Our description is based on the geometrization of the quantum state space by introducing an information metric which defines a non-unique measure of distinguishability on the state space. We show in particular how our approach incorporates and unifies the previous specialized results, interpreting them under a new comprehensive framework, and allowing us to reach significantly beyond. From the physical point of view, our investigation is the first to highlight the role of populations versus quantum coherences in the determination and saturation of the speed limits. Our results can find applications in the optimization of quantum protocols in quantum computation and metrology, and might provide new insights in fundamental investigations of quantum thermodynamics.

As últimas décadas testemunharam intensa atividade de pesquisa teórica e experimental visando compreender o conceito do tempo na mecânica quântica. Este tema desencadeou significante progresso na busca por dispositivos mais rápidos e eficientes no processamento de informação e implementação de tecnologias de comunicação. Motivados pela pergunta quão rápido um sistema quântico evolui sob uma dada dinâmica?, tais avanços levaram a formulação do chamado limite quântico de velocidade ou quantum speed limit, (QSL), i.e., um limite inferior definindo o tempo mínimo de evolução entre estados quânticos distintos. Diversos resultados reportaram QSLs obtidos via tratamentos diferentes e aparentemente desconexos, muitas vezes sob configurações específicas, que deixaram uma lacuna fundamental à resposta da questão geral colocada anteriormente. Neste projeto investigamos como a não-unicidade de uma medida de distinguibilidade de operadores densidade definida no espaço de estados quânticos influencia o QSL e pode ser explorada no intuito de obter limites inferiores mais robustos no tempo de evolução de estados arbitrários. Em particular, baseando-nos no formalismo da geometria da informação, estabelecemos uma família infinita de QSLs válidos para evoluções unitárias e não-unitárias. Este trabalho se propõe unificar e generalizar resultados existentes sobre QSLs na literatura, além de fornecer exemplos de limites mais precisos do que aqueles baseados na informação de Fisher convencional. Em termos físicos, esta investigação é a primeira a destacar o papel das populações e coerências quânticas no cálculo e saturação dos QSLs. Nossos resultados podem encontrar aplicações na otimização de protocolos em computação quântica e metrologia, além de fornecer novos pontos de vista em investigações fundamentais da termodinâmica quântica. / The last decades witnessed intense theoretical and experimental research activity in order to understand the concept of time in quantum mechanics. This subject triggered significant progress in the search for faster and efficient schemes in the implementation of quantum information and communication technologies. Starting from the puzzle How fast can a quantum state evolve under a given dynamics?, such advances have led to the establishment of quantum speed limits (QSLs), i.e., a lower bound setting the minimum time evolution between two distinct quantum states. Past results have included different, apparently unrelated approaches to quantum speed limits, and sometimes tailored to specific settings, which therefore left a fundamental gap in obtaining a satisfactory answer to the general question posed above. In this work we provide a breakthrough for the study and applications of quantum speed limits. We approach the problem from a general information theoretic point of view and we adopt an elegant geometric formalism to construct an infinite family of quantum speed limits valid for closed and open system evolutions. Our description is based on the geometrization of the quantum state space by introducing an information metric which defines a non-unique measure of distinguishability on the state space. We show in particular how our approach incorporates and unifies the previous specialized results, interpreting them under a new comprehensive framework, and allowing us to reach significantly beyond. From the physical point of view, our investigation is the first to highlight the role of populations versus quantum coherences in the determination and saturation of the speed limits. Our results can find applications in the optimization of quantum protocols in quantum computation and metrology, and might provide new insights in fundamental investigations of quantum thermodynamics.

京都大学 / 新制・課程博士 / 博士(医科学) / 甲第23108号 / 医科博第119号 / 新制||医科||8(附属図書館) / 京都大学大学院医学研究科医科学専攻 / (主査)教授 藤渕 航, 教授 松田 道行, 教授 黒田 知宏 / 学位規則第4条第1項該当 / Doctor of Medical Science / Kyoto University / DFAM

In this dissertation, I consider the problem of model reduction in both oscillatory and networked systems. Previously, the Manifold Boundary Approximation Method (MBAM) has been demonstrated as a data-driven tool for reducing the parametric complexity of so-called sloppy models. To be effective, MBAM requires the model manifold to have low curvature. I show that oscillatory models are characterized by model manifolds with high curvature in one or more directions. I propose methods for transforming the model manifolds of these models into ones with low curvature and demonstrate on a couple of test systems. I demonstrate MBAM as a tool for data-driven network reduction on a small model from power systems. I derive multiple effective networks for the model, each tailored to a specific choice of system observations. I find several important types of parameter reductions, including network reductions, which can be used in large power systems models. Finally, I consider the problem of piecemeal reduction of large systems. When a large system is split into pieces that are to be reduced separately using MBAM, there is no guarantee that the reduced pieces will be compatible for reassembly. I propose a strategy for reducing a system piecemeal while guaranteeing that the reduced pieces will be compatible. I demonstrate the reduction strategy on a small resistor network.

Genome-wide studies of diseases and chronic conditions frequently fail to uncover marked or consistent differences in RNA or protein concentrations. However, the developing field of kinetic proteomics has made promising discoveries in differences in the turnover rate of these same proteins, even when concentrations were not necessarily different. The situation can theoretically be modeled mathematically using bifurcation equations, but uncovering the proper form of these is difficult. To this end, we developed TWIG, a method for characterizing bifurcations that leverages information geometry to identify drivers of complex systems. Using this, we characterized the bifurcation and stability properties of all 132 possible 3- and 22,662 possible 4-node subgraphs (motifs) of protein-protein interaction networks. Analyzing millions of real world protein networks indicates that natural selection has little preference for motifs that are stable per se, but a great preference for motifs who have parameter regions that are exclusively stable, rather than poorly constrained mixtures of stability and instability. We apply this knowledge to mice on calorie restricted (CR) diets, demonstrating that changes in their protein turnover rates do indeed make their protein networks more stable, explaining why CR is the most robust way known to extend lifespan.

Uncertainty quantification (UQ) is an increasingly important part of materials modeling. In this paper, we consider the problem of quantifying parametric uncertainty in classical empirical interatomic potentials (IPs). Previous work based on local sensitivity analysis using the Fisher Information has shown that IPs are sloppy, i.e., are insensitive to coordinated changes of many parameter combinations. We confirm these results and further explore the non-local statistics in the context of sloppy model analysis using both Bayesian (MCMC) and Frequentist (profile likelihood) methods. We interface these tools with the Knowledgebase of Interatomic Models (OpenKIM) and study three models based on the Lennard-Jones, Morse, and Stillinger-Weber potentials, respectively. We confirm that IPs have global properties similar to those of sloppy models from fields such as systems biology, power systems, and critical phenomena. These models exhibit a low effective dimensionality in which many of the parameters are unidentifiable, i.e., do not encode any information when fit to data. Because the inverse problem in such models is ill-conditioned, unidentifiable parameters present challenges for traditional statistical methods. In the Bayesian approach, Monte Carlo samples can depend on the choice of prior in subtle ways. In particular, they often "evaporate" parameters into high-entropy, sub-optimal regions of the parameter space. For profile likelihoods, confidence regions are extremely sensitive to the choice of confidence level. To get a better picture of the relationship between data and parametric uncertainty, we sample the Bayesian posterior at several sampling temperatures and compare the results with those of Frequentist analyses. In analogy to statistical mechanics, we classify samples as either energy-dominated, i.e., characterized by identifiable parameters in constrained (ground state) regions of parameter space, or entropy-dominated, i.e., characterized by unidentifiable (evaporated) parameters. We complement these two pictures with information geometry to illuminate the underlying cause of this phenomenon. In this approach, a parameterized model is interpreted as a manifold embedded in the space of possible data with parameters as coordinates. We calculate geodesics on the model manifold and find that IPs, like other sloppy models, have bounded manifolds with a hierarchy of widths, leading to low effective dimensionality in the model. We show how information geometry can motivate new, natural parameterizations that improve the stability and interpretation of UQ analysis and further suggest simplified, less-sloppy models.

The aim of this work is to lay down a formalism for parametric models that encapsulates both Classical and Quantum Information Geometry. This will be done introducing parametric models on spaces of normal positive linear functionals on W*-algebras and providing a way of defining a Riemannian structure on this models that comes from the Jordan product of the W*-algebra. This Riemannian structure will have some features that are appealing from the viewpoint of Information Geometry. After introducing this W*-algebraic framework, we will move to Estimation Theory. We will see how and to what extent it is possible to formulate in this framework two well-known statistical bounds: the Cramér-Rao bound and the Helstrom bound. Finally, we will explicitly construct some examples that show how it is possible to reduce this general framework to obtain well-known structures in Classical and Quantum Information Geometry.

In this thesis we aim to develop new perspectives on the statistical mechanics of black holes using an information geometric approach (Ruppeiner and Weinhold geometry). The Ruppeiner metric is defined as a Hessian matrix on a Gibbs surface, and provides a geometric description of thermodynamic systems in equilibrium. This Ruppeiner geometry exhibits physically suggestive features; a flat Ruppeiner metric for systems with no interactions i.e. the ideal gas, and curvature singularities signaling critical behavior(s) of the system. We construct a flatness theorem based on the scaling property of the black holes, which proves to be useful in many cases. Another thermodynamic geometry known as the Weinhold geometry is defined as the Hessian of internal energy and is conformally related to the Ruppeiner metric with the system’s temperature as a conformal factor.  We investigate a number of black hole families in various gravity theories. Our findings are briefly summarized as follows: the Reissner-Nordström type, the Einstein-Maxwell-dilaton andBTZ black holes have flat Ruppeiner metrics that can be represented by a unique state space diagram. We conjecture that the state space diagram encodes extremality properties of the black hole solution. The Kerr type black holes have curved Ruppeiner metrics whose curvature singularities are meaningful in five dimensions and higher, signifying the onset of thermodynamic instabilities of the black hole in higher dimensions. All the three-parameter black hole families in our study have non-flat Ruppeiner and Weinhold metrics and their associated curvature singularities occur in the extremal limits. We also study two-dimensional black hole families whose thermodynamic geometries are dependent on parameters that determine the thermodynamics of the black hole in question. The tidal charged black hole which arises in the braneworld gravity is studied. Despite its similarity to the Reissner-Nordström type, its thermodynamic geometries are distinctive. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Submitted. / Geometry and Physics

Nous nous intéressons à la comparaison de formes de courbes lisses prenant leurs valeurs dans une variété riemannienne M. Dans ce but, nous introduisons une métrique riemannienne invariante par reparamétrisations sur la variété de dimension infinie des immersions lisses dans M. L’équation géodésique est donnée et les géodésiques entre deux courbes sont construites par tir géodésique. La structure quotient induite par l’action du groupe des reparamétrisations sur l’espace des courbes est étudiée. À l’aide d’une décomposition canonique d’un chemin dans un fibré principal, nous proposons un algorithme qui construit la géodésique horizontale entre deux courbes et qui fournit un matching optimal. Dans un deuxième temps, nous introduisons une discrétisation de notre modèle qui est elle-même une structure riemannienne sur la variété de dimension finie Mn+1 des "courbes discrètes" définies par n + 1 points, où M est de courbure sectionnelle constante. Nous montrons la convergence du modèle discret vers le modèle continu, et nous étudions la géométrie induite. Des résultats de simulations dans la sphère, le plan et le demi-plan hyperbolique sont donnés. Enfin, nous donnons le contexte mathématique nécessaire à l’application de l’étude de formes dans une variété au traitement statistique du signal radar, où des signaux radars localement stationnaires sont représentés par des courbes dans le polydisque de Poincaré via la géométrie de l’information. / We are concerned with the comparison of the shapes of open smooth curves that take their values in a Riemannian manifold M. To this end, we introduce a reparameterization invariant Riemannian metric on the infinite-dimensional manifold of these curves, modeled by smooth immersions in M. We derive the geodesic equation and solve the boundary value problem using geodesic shooting. The quotient structure induced by the action of the reparametrization group on the space of curves is studied. Using a canonical decomposition of a path in a principal bundle, we propose an algorithm that computes the horizontal geodesic between two curves and yields an optimal matching. In a second step, restricting to base manifolds of constant sectional curvature, we introduce a detailed discretization of the Riemannian structure on the space of smooth curves, which is itself a Riemannian metric on the finite-dimensional manifold Mn+1 of "discrete curves" given by n + 1 points. We show the convergence of the discrete model to the continuous model, and study the induced geometry. We show results of simulations in the sphere, the plane, and the hyperbolic halfplane. Finally, we give the necessary framework to apply shape analysis of manifold-valued curves to radar signal processing, where locally stationary radar signals are represented by curves in the Poincaré polydisk using information geometry.