Distances in evidence theory: Comprehensive survey and generalizations

AbstractThe purpose of the present work is to survey the dissimilarity measures defined so far in the mathematical framework of evidence theory, and to propose a classification of these measures based on their formal properties. This research is motivated by the fact that while dissimilarity measures have been widely studied and surveyed in the fields of probability theory and fuzzy set theory, no comprehensive survey is yet available for evidence theory. The main results presented herein include a synthesis of the properties of the measures defined so far in the scientific literature; the generalizations proposed naturally lead to additions to the body of the previously known measures, leading to the definition of numerous new measures. Building on this analysis, we have highlighted the fact that Dempster’s conflict cannot be considered as a genuine dissimilarity measure between two belief functions and have proposed an alternative based on a cosine function. Other original results include the justification of the use of two-dimensional indexes as (cosine; distance) couples and a general formulation for this class of new indexes. We base our exposition on a geometrical interpretation of evidence theory and show that most of the dissimilarity measures so far published are based on inner products, in some cases degenerated. Experimental results based on Monte Carlo simulations illustrate interesting relationships between existing measures

Positive definite indistinguishability operators

Peer ReviewedPostprint (author's final draft

Articles indexats publicats per autors de l'ETSAB

Aquest document recull els articles publicats per investigadors de l'ETSAB en revistes del Web of Science i de Scopus des de l'any 2000 fins el 2011.Preprin

Networked Data Analytics: Network Comparison And Applied Graph Signal Processing

Networked data structures has been getting big, ubiquitous, and pervasive. As our day-to-day activities become more incorporated with and influenced by the digital world, we rely more on our intuition to provide us a high-level idea and subconscious understanding of the encountered data. This thesis aims at translating the qualitative intuitions we have about networked data into quantitative and formal tools by designing rigorous yet reasonable algorithms. In a nutshell, this thesis constructs models to compare and cluster networked data, to simplify a complicated networked structure, and to formalize the notion of smoothness and variation for domain-specific signals on a network. This thesis consists of two interrelated thrusts which explore both the scenarios where networks have intrinsic value and are themselves the object of study, and where the interest is for signals defined on top of the networks, so we leverage the information in the network to analyze the signals. Our results suggest that the intuition we have in analyzing huge data can be transformed into rigorous algorithms, and often the intuition results in superior performance, new observations, better complexity, and/or bridging two commonly implemented methods. Even though different in the principles they investigate, both thrusts are constructed on what we think as a contemporary alternation in data analytics: from building an algorithm then understanding it to having an intuition then building an algorithm around it. We show that in order to formalize the intuitive idea to measure the difference between a pair of networks of arbitrary sizes, we could design two algorithms based on the intuition to find mappings between the node sets or to map one network into the subset of another network. Such methods also lead to a clustering algorithm to categorize networked data structures. Besides, we could define the notion of frequencies of a given network by ordering features in the network according to how important they are to the overall information conveyed by the network. These proposed algorithms succeed in comparing collaboration histories of researchers, clustering research communities via their publication patterns, categorizing moving objects from uncertain measurmenets, and separating networks constructed from different processes. In the context of data analytics on top of networks, we design domain-specific tools by leveraging the recent advances in graph signal processing, which formalizes the intuitive notion of smoothness and variation of signals defined on top of networked structures, and generalizes conventional Fourier analysis to the graph domain. In specific, we show how these tools can be used to better classify the cancer subtypes by considering genetic profiles as signals on top of gene-to-gene interaction networks, to gain new insights to explain the difference between human beings in learning new tasks and switching attentions by considering brain activities as signals on top of brain connectivity networks, as well as to demonstrate how common methods in rating prediction are special graph filters and to base on this observation to design novel recommendation system algorithms

Multi-Modal Similarity Learning for 3D Deformable Registration of Medical Images

Alors que la perspective de la fusion d images médicales capturées par des systèmes d imageries de type différent est largement contemplée, la mise en pratique est toujours victime d un obstacle théorique : la définition d une mesure de similarité entre les images. Des efforts dans le domaine ont rencontrés un certain succès pour certains types d images, cependant la définition d un critère de similarité entre les images quelle que soit leur origine et un des plus gros défis en recalage d images déformables. Dans cette thèse, nous avons décidé de développer une approche générique pour la comparaison de deux types de modalités donnés. Les récentes avancées en apprentissage statistique (Machine Learning) nous ont permis de développer des solutions innovantes pour la résolution de ce problème complexe. Pour appréhender le problème de la comparaison de données incommensurables, nous avons choisi de le regarder comme un problème de plongement de données : chacun des jeux de données est plongé dans un espace commun dans lequel les comparaisons sont possibles. A ces fins, nous avons exploré la projection d un espace de données image sur l espace de données lié à la seconde image et aussi la projection des deux espaces de données dans un troisième espace commun dans lequel les calculs sont conduits. Ceci a été entrepris grâce à l étude des correspondances entre les images dans une base de données images pré-alignées. Dans la poursuite de ces buts, de nouvelles méthodes ont été développées que ce soit pour la régression d images ou pour l apprentissage de métrique multimodale. Les similarités apprises résultantes sont alors incorporées dans une méthode plus globale de recalage basée sur l optimisation discrète qui diminue le besoin d un critère différentiable pour la recherche de solution. Enfin nous explorons une méthode qui permet d éviter le besoin d une base de données pré-alignées en demandant seulement des données annotées (segmentations) par un spécialiste. De nombreuses expériences sont conduites sur deux bases de données complexes (Images d IRM pré-alignées et Images TEP/Scanner) dans le but de justifier les directions prises par nos approches.Even though the prospect of fusing images issued by different medical imagery systems is highly contemplated, the practical instantiation of it is subject to a theoretical hurdle: the definition of a similarity between images. Efforts in this field have proved successful for select pairs of images; however defining a suitable similarity between images regardless of their origin is one of the biggest challenges in deformable registration. In this thesis, we chose to develop generic approaches that allow the comparison of any two given modality. The recent advances in Machine Learning permitted us to provide innovative solutions to this very challenging problem. To tackle the problem of comparing incommensurable data we chose to view it as a data embedding problem where one embeds all the data in a common space in which comparison is possible. To this end, we explored the projection of one image space onto the image space of the other as well as the projection of both image spaces onto a common image space in which the comparison calculations are conducted. This was done by the study of the correspondences between image features in a pre-aligned dataset. In the pursuit of these goals, new methods for image regression as well as multi-modal metric learning methods were developed. The resulting learned similarities are then incorporated into a discrete optimization framework that mitigates the need for a differentiable criterion. Lastly we investigate on a new method that discards the constraint of a database of images that are pre-aligned, only requiring data annotated (segmented) by a physician. Experiments are conducted on two challenging medical images data-sets (Pre-Aligned MRI images and PET/CT images) to justify the benefits of our approach.CHATENAY MALABRY-Ecole centrale (920192301) / SudocSudocFranceF

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Die Dichte links-invarianter Metriken auf kompakten zusammenhängenden Liegruppen

The subject of the present thesis is to study the thickness of left-invariant riemannian metrics Q on compact connected d-dimensional Lie groups G with respect to the existence of globally or locally thinnest metrics. This is a variation of a problem posed by Berger concerning the identification of particularly good riemannian metrics on compact manifolds. If the group in question is abelian, the question of the existence of a thinnest metric may be answered in the affirmative. By contrast, the existence of arbitrarily thin metrics on all non-abelian groups is proved using the theory of Carnot–Carathéodory metrics. Furthermore, it is shown that the bi-invariant metrics of certain simple Lie groups with finitely many antipodes may be continuously deformed into thinner metrics by shrinking a maximal torus. The affected class of Lie groups consists of the simply-connected Lie groups of the infinite families A n , B n and D n , the simply-connected exceptional Lie groups E 6 and E 7 as well as SO(2n) for n ≥ 2. The existence of locally thinnest metrics on these groups remains unsettled. Finally, it is proved that the bi-invariant metrics of the Lie group PSU(n + 1) for n ≥ 1 are locally optimal metrics, meaning that their thickness does not decrease under any deformations except (possibly) those contained in a certain zero set. It remains unknown whether the bi-invariant metrics are locally thinnest metrics, except in the affirmative case of PSU(2). The problematic zero set is described as a subset of deformations that preserve the volume of every maximal torus up to first order, or, equivalently, deformations whose inertia operator is perpendicular to the second Cartan power of the Lie algebra.Gegenstand der vorliegenden Dissertationsschrift ist die Untersuchung der Dichte links-invarianter riemannscher Metriken Q auf kompakten zusammenhängenden d-dimensionalen Liegruppen G hinsichtlich der Existenz global oder lokal dünnster Metriken. Es handelt sich dabei um eine Variante des von Berger formulierten Problems über die Bestimmung besonders gutartiger riemannscher Metriken auf kompakten Mannigfaltigkeiten. Liegt eine abelsche Gruppe vor, so ist die Frage nach der Existenz einer dünnsten Metrik positiv zu beantworten. Im Gegensatz dazu wird die Existenz von beliebig dünnen Metriken auf allen nicht-abelschen Liegruppen mit Hilfe der Theorie der Carnot–Carathéodory-Metriken nachgewiesen. Darüber hinaus wird gezeigt, dass die bi-invarianten Metriken bestimmter einfacher Liegruppen mit endlich vielen Antipoden durch das Zusammenziehen eines maximalen Torus in stetiger Weise zu dünneren Metriken verformt werden können. Die betroffene Klasse von Liegruppen besteht aus den einfach-zusammenhängenden Liegruppen der unendlichen Familien A n , C n und D n , den einfach-zusammenhängenden exzeptionellen Liegruppen E 6 und E 7 sowie SO(2n) für n ≥ 2. Die Existenz lokal dünnster Metriken auf diesen Gruppen bleibt weiter ungeklärt. Schließlich wird bewiesen, dass die bi-invarianten Metriken der Liegruppe PSU(n + 1) für n ≥ 1 lokal optimale Metriken sind, was bedeutet, dass ihre Dichte unter keiner Deformation kleiner wird, mit der möglichen Ausnahme solcher Deformationen, die in einer bestimmten Nullmenge enthalten sind. Die Frage, ob die bi-invarianten Metriken lokal dünnste Metriken sind, bleibt unbeantwortet, außer im Falle von PSU(2), in dem sie positiv beantwortet werden kann. Die problematische Nullmenge lässt sich beschreiben als Teilmenge der Deformationen, die in erster Näherung das Volumen keines maximalen Tori ändern, bzw. deren Trägheitsoperatoren senkrecht zur zweiten Cartanpotenz der Liealgebra stehen