Consistance des statistiques dans les espaces quotients de dimension infinie

In computational anatomy, organ shapes are assumed to be deformation of a common template. The data can be organ images but also organ surfaces, and the deformations are often assumed to be diffeomorphisms. In order to estimate the template, one often uses the max-max algorithm which minimizes, among all the prospective templates, the sum of the squared distance after registration between the data and a prospective template. Registration is here the step of the algorithm which finds the best deformation between two shapes. The goal of this thesis is to study this template estimation method from a mathematically point of view. We prove in particular that this algorithm is inconsistent due to the noise. This means that even with an infinite number of data, and with a perfect minimization algorithm, one estimates the original template with an error. In order to prove inconsistency, we formalize the template estimation: deformations are assumed to be random elements of a group which acts on the space of observations. Besides, the studied algorithm is interpreted as the computation of the Fréchet mean in the space of observations quotiented by the group of deformations. In this thesis, we prove that the inconsistency comes from the contraction of the distance in the quotient space with respect to the distance in the space of observations. Besides, we obtained a Taylor expansion of the consistency bias with respect to the noise level. As a consequence, the inconsistency is unavoidable when the noise level is high.En anatomie computationnelle, on suppose que les formes d'organes sont issues des déformations d'un template commun. Les données peuvent être des images ou des surfaces d'organes, les déformations peuvent être des difféomorphismes. Pour estimer le template, on utilise souvent un algorithme appelé «max-max» qui minimise parmi tous les candidats, la somme des carrées des distances après recalage entre les données et le template candidat. Le recalage est l'étape de l'algorithme qui trouve la meilleure déformation pour passer d'une forme à une autre. Le but de cette thèse est d'étudier cet algorithme max-max d'un point de vue mathématique. En particulier, on prouve que cet algorithme est inconsistant à cause du bruit. Cela signifie que même avec un nombre infini de données et avec un algorithme de minimisation parfait, on estime le template original avec une erreur non nulle. Pour prouver l'inconsistance, on formalise l'estimation du template. On suppose que les déformations sont des éléments aléatoires d'un groupe qui agit sur l'espace des observations. L'algorithme étudié est interprété comme le calcul de la moyenne de Fréchet dans l'espace des observations quotienté par le groupe des déformations. Dans cette thèse, on prouve que l'inconsistance est dû à la contraction de la distance quotient par rapport à la distance dans l'espace des observations. De plus, on obtient un équivalent de biais de consistance en fonction du niveau de bruit. Ainsi, l'inconsistance est inévitable quand le niveau de bruit est suffisamment grand

Proceedings of the fifth international workshop on Mathematical Foundations of Computational Anatomy (MFCA 2015)

International audienceComputational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information.The Mathematical Foundations of Computational Anatomy (MFCA) workshop aims at fostering the interactions between the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum for the exchange of the theoretical ideas and aims at being a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations.Following the first edition of this workshop in 20061, the second edition in New-York in 20082, the third edition in Toronto in 20113, the forth edition in Nagoya Japan on September 22 20134, the fifth edition was held in Munich on October 9 20155.Contributions were solicited in Riemannian, sub-Riemannian and group theoretical methods, advanced statistics on deformations and shapes, metrics for computational anatomy, statistics of surfaces, time-evolving geometric processes, stratified spaces, optimal transport, approximation methods in statistical learning and related subjects. Among the submitted papers, 14 were selected andorganized in 4 oral sessions

Fréchet means in Wasserstein space:theory and algorithms

This work studies the problem of statistical inference for Fréchet means in the Wasserstein space of measures on Euclidean spaces, $\mathcal W_2 ( \mathbb R^d )$. This question arises naturally from the problem of separating amplitude and phase variation in point processes, analogous to a well-known problem in functional data analysis. We formulate the point process version of the problem, show that it is canonically equivalent to that of estimating Fréchet means in $\mathcal W_2 ( \mathbb R d )$, and carry out estimation by means of $M$-estimation. This approach allows to achieve consistency in a genuinely nonparametric framework, even in a sparse sampling regime. For Cox processes on the real line, consistency is supplemented by convergence rates and, in the dense sampling regime, $\sqrt n$-consistency and a central limit theorem. Computation of the Fréchet mean is challenging when the processes are multivariate, in which case our Fréchet mean estimator is only defined implicitly as the minimiser of an optimisation problem. To overcome this difficulty, we propose a steepest descent algorithm that approximates the minimiser, and show that it converges to a local minimum. Our techniques are specific to the Wasserstein space, because Hessian-type arguments that are commonly used for similar convergence proofs do not apply to that space. In addition, we discuss similarities with generalised Procrustes analysis. The key advantage of the algorithm is that it requires only the solution of pairwise transportation problems. The results in the preceding paragraphs require properties of Fréchet means in $\mathcal W_2 ( \mathbb R ^d )$ whose theory is developed, supplemented by some new results. We present the tangent bundle and exploit its relation to optimal maps in order to derive differentiability properties of the associated Fréchet functional, obtaining a characterisation of Karcher means. Additionally, we establish a new optimality criterion for local minima and prove a new stability result for the optimal maps that, enhanced with the established consistency of the Fréchet mean estimator, yields consistency of the optimal transportation maps

Functional data analysis: interpolation, registration, and nearest neighbors in scikit-fda

Functional Data Analysis (FDA) is a branch of Statistics devoted to the study of random quantities that depend on a continuous parameter, such as time series or curves in space. In FDA the data instances can be viewed as random functions sampled from an underlying stochastic process. In this work we consider three different tasks in FDA: the use of interpolation techniques to estimate the values of the functions at unobserved points, the registration of these type of data, and the solution of classification and regression problems in which the instances are characterized by functional attributes. In particular, in this project the scikit-fda package for FDA in Python has been extended with functionality in these areas. Generally, the data instances considered in FDA consist of a collection of observations at a discrete values of the parameter on which they depend (e.g. time or space). For some applications it is convenient, and in some cases necessary, to estimate the value of these functions at unobserved points. This can be achieved through the use of interpolation from the available measurements. In some applications, the functions observed have similar shapes, but exhibit variability whose origin can be traced to distortions in the scale of the continuous parameter on which the data depend. Registration consists in characterizing this variability and eliminating it from the sample considered. In this work we also address classification and regression problems with data that are characterized by functions. Specifically, we design nearest neighbors estimators based on the notion of closeness among samples. Specifically, in this work the scikit-fda package has been extended to include interpolation methods based on splines. The package has also been endowed with tools for data registration using either shifts, landmark alignment, or elastic registration, which makes use of the Fisher-Rao metric to align the functions in a sample. In addition, models based on nearest neighbors have been included to carry out regression, with both scalar and functional response, and classification

Analyzing Mathematicians\u27 Concept Images of Differentials

The differential is a symbol that is common in first- and second-year calculus. It is perhaps expected that a common mathematical symbol would be interpreted universally. However, recent literature that addresses student interpretations of differentials, usually in the context of definite integration, suggests that this is not the case, and that many interpretations are possible. Reviews of textbooks showed that there was not a lot of discussion about differentials, and what interpretations there were depended upon the context in which the differentials were presented. This dissertation explores some of these issues. Since students may not have the experience necessary to build their own interpretations totally free of their instructors’ influences, I chose to interview experienced mathematicians for their differential interpretations. Most of the current literature involves the differential within the context of definite integrals; my work expands on this literature by exploring additional expressions that contain differentials. The goal was to build a dataset of multiple instructors’ interpretations of multiple differentials to see how uniform those interpretations were. Initial interviews discussing five expressions which contained differentials, three contexts in which most of these expressions were used, and auxiliary questions that asked the meaning of “differential,” the differences between and , and the interpretation of phrases used to describe infinitely small quantities were conducted with seven expert mathematicians from a large research university. By analyzing the responses given by these mathematicians, two lists of themes were created: one based on remarks that address the quality of the differential directly, and one based on remarks that address one’s feelings about differentials. In addition, for the responses that address differentials directly, a flowchart was created to guide each of these responses to its proper theme. After the creation of these lists, three more mathematicians were interviewed to ensure that the theme lists would still be valid outside of the interviews used to create them. Not only was no overall formal concept image for the differential found, but many different and sometimes contrasting themes were found within each interview subject’s personal concept image. A framework for categorizing the multiple conceptualizations that were found for the differentials themselves was created, as well as a beginning list of ancillary themes that address possible thoughts about and uses of differentials. The dissertation concludes with a list of possible teaching implications that might arise from the existence of multiple differential conceptualizations, as well as some suggested future research that might expand upon this work

An Invitation to Statistics in Wasserstein Space

This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph. ; Gives a succinct introduction to necessary mathematical background, focusing on the results useful for statistics from an otherwise vast mathematical literature. Presents an up to date overview of the state of the art, including some original results, and discusses open problems. Suitable for self-study or to be used as a graduate level course text. Open access

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum