Multidimensional Interleavings and Applications to Topological Inference

This work concerns the theoretical foundations of persistence-based topological data analysis. We develop theory of topological inference in the multidimensional persistence setting, and directly at the (topological) level of filtrations rather than only at the (algebraic) level of persistent homology modules. Our main mathematical objects of study are interleavings. These are tools for quantifying the similarity between two multidimensional filtrations or persistence modules. They were introduced for 1-D filtrations and persistence modules by Chazal, Cohen-Steiner, Glisse, Guibas, and Oudot. We introduce generalizations of the definitions of interleavings given by Chazal et al. and use these to define pseudometrics, called interleaving distances, on multidimensional filtrations and multidimensional persistence modules. We present an in-depth study of interleavings and interleaving distances. We then use them to formulate and prove several multidimensional analogues of a topological inference theorem of Chazal, Guibas, Oudot, and Skraba. These results hold directly at the level of filtrations; they yield as corollaries corresponding results at the module level.Comment: Late stage draft of Ph.D. thesis. 176 pages. Expands upon content in arXiv:1106.530

Analyse structurelle de l'hydrogène neutre dans la voie lactée

Les étoiles vivent et meurent en rejetant de la matière dans le milieu interstellaire (MIS) et elles naissent à l’intérieur de celui-ci. Nous avons analysé la composante d’hydrogène neutre du MIS. Nos données proviennent de la partie canadienne de l’International Galactic Plane Survey qui vise l’imagerie spectroscopique de l’hydrogène neutre du plan de notre galaxie. Nous avons utilisé deux outils mathématiques d’analyse d’images: la technique d’Espaces Métriques (TEM) et la méthode des Maxima du Module de la Transformée en Ondelettes (MMTO). La TEM est un formalisme mathématique d’analyse d’images qui permet de comparer quantitativement la complexité des objets étudiés. Nous avons amélioreré l’outil aux niveaux mathématique et technique avant de l’utiliser pour caractériser la complexité de 28 régions d’hydrogéne neutre. Aprés avoir classé les 28 objets, nous avons trouvé des corrélations entre ce classement et les propriétés physiques des objets sous-jacents, dont: (1) Plus le flux des photons UV est élevé, plus la région de H i photodissociée est complexe; et (2) la complexité des régions H i augmente avec l’ˆage des restes de supernovae auxquels elles sont associées. La méthode MMTO est un formalisme multifractal basé sur la transformée en ondelettes. Nos résultats obtenus à partir de cette méthode concernent les propriétés multifractales et anisotropes de l’hydrogène neutre dans notre galaxie. Les nuages terrestres exhibent des propriétés multifractales. Nous avons démontré que l’hydrogène neutre du disque de notre galaxie est monofractal. En analysant séparément les bras spiraux et les milieux inter-bras, nous avons découvert une signature anisotrope et que les structures horizontales sont plus complexes que les structures verticales. Cette anisotropie est indépendante de l’échelle pour les inter-bras tandis qu’elle est dépendante de l’échelle pour les bras spiraux. Les hypothèses investiguées pour obtenir une explication physique sont: le gradient de distribution en z (“scale-height gradient”), l’onde de densité, l’activité de formation d’étoiles, la photo-lévitation de nuages poussiéreux, les mouvements aléatoires de nuages H i, la corrugation et la turbulence.Stars live and die by rejecting matter in the interstellar medium (ISM), where they were born. We have analyzed the neutral hydrogen component of the ISM. The data come from the Canadian portion of the International Galactic Plane Survey which aims the spectroscopic imaging of the neutral hydrogen from our Galaxy. We have used two mathematical image analysis tools: Metric Space Technique (MST) and the Wavelet Transform Modulus Maxima (WTMM) method. The MST is an image analysis mathematical formalism that allows one to quantitatively compare the complexity of the studied objects. We have improved the tool mathematically and technically before using it to characterize the complexity of 28 neutral hydrogen regions. After classifying the 28 objects, we have found some correlations between this ranking and the physical properties of the underlying objects, for example: (1) The complexity of the photodissociated neutral hydrogen regions increases with the flux of UV photons; and (2) the complexity of neutral hydrogen regions increases with the age of the supernovae remnants to which they are associated. The WTMM method is a multifractal formalism based on the wavelet transform. The results we obtain from this method concern the multifractal and anisotropic properties of neutral hydrogen in our Galaxy. Earth clouds exhibit multifractal properties. We have shown that the neutral hydrogen from our galactic disk is monofractal. By analyzing separately spiral arms and the inter-arm regions, we have discovered an anisotropic signature and that the horizontal structures and more complex than the vertical structures. This anisotropy is independent of scale for the inter-arms while it is depedent of scale for the spiral arms. The investigated hypotheses to obtain some physical explanations are: the scale-height gradient, the density wave, star formation activity, photo-levitation of dusty clouds, random motions of neutral hydrogen clouds, corrugation and turbulence

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

Topics in shape-constrained inference

This thesis consists of three chapters. The first of these surveys the field of nonparametric inference under shape constraints, focussing in particular on the topics of shape-restricted regression and shape-constrained density estimation. In the second chapter, we investigate the adaptation properties of the log-concave maximum likelihood estimator of a multivariate log-concave density. Our main theoretical results demonstrate that in certain situations where the true density has additional structure, the estimator can attain rates of convergence (with respect to squared Hellinger distance or Kullback-Leibler divergence) that are strictly faster than the global minimax convergence rate. We illustrate three different types of adaptive behaviour in dimensions 2 and 3 through sharp oracle inequalities. Our approach entails developing local bracketing entropy bounds for Hellinger neighbourhoods of log-concave densities that belong to the special subclasses described above. To this end, we apply techniques from convex geometry and real analysis to elucidate the structural properties of such densities, and obtain some results of independent interest. In the third chapter, we consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimisation problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a general projection framework that demonstrates the consistency and robustness to misspecification of the estimator, we prove worst-case and adaptive risk bounds for the estimation of the regression function, in the form of sharp oracle inequalities, and establish bounds on the rate of convergence of the estimated inflection point. These theoretical results reveal not only that the estimator achieves the minimax optimal rate for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations also confirm the desirable finite-sample properties of the estimator, and our algorithm is implemented in the R package 'Sshaped'