34 research outputs found
On minimax density estimation via measure transport
We study the convergence properties, in Hellinger and related distances, of
nonparametric density estimators based on measure transport. These estimators
represent the measure of interest as the pushforward of a chosen reference
distribution under a transport map, where the map is chosen via a maximum
likelihood objective (equivalently, minimizing an empirical Kullback-Leibler
loss) or a penalized version thereof. We establish concentration inequalities
for a general class of penalized measure transport estimators, by combining
techniques from M-estimation with analytical properties of the transport-based
density representation. We then demonstrate the implications of our theory for
the case of triangular Knothe-Rosenblatt (KR) transports on the -dimensional
unit cube, and show that both penalized and unpenalized versions of such
estimators achieve minimax optimal convergence rates over H\"older classes of
densities. Specifically, we establish optimal rates for unpenalized
nonparametric maximum likelihood estimation over bounded H\"older-type balls,
and then for certain Sobolev-penalized estimators and sieved wavelet
estimators.Comment: 27 page
Inference via low-dimensional couplings
We investigate the low-dimensional structure of deterministic transformations
between random variables, i.e., transport maps between probability measures. In
the context of statistics and machine learning, these transformations can be
used to couple a tractable "reference" measure (e.g., a standard Gaussian) with
a target measure of interest. Direct simulation from the desired measure can
then be achieved by pushing forward reference samples through the map. Yet
characterizing such a map---e.g., representing and evaluating it---grows
challenging in high dimensions. The central contribution of this paper is to
establish a link between the Markov properties of the target measure and the
existence of low-dimensional couplings, induced by transport maps that are
sparse and/or decomposable. Our analysis not only facilitates the construction
of transformations in high-dimensional settings, but also suggests new
inference methodologies for continuous non-Gaussian graphical models. For
instance, in the context of nonlinear state-space models, we describe new
variational algorithms for filtering, smoothing, and sequential parameter
inference. These algorithms can be understood as the natural
generalization---to the non-Gaussian case---of the square-root
Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure
Méthodes unidimensionnelles et d'évolution pour le transport optimal
Sur une droite, le transport optimal ne pose pas de difficultés. Récemment, ce constat a été utilisé pour traiter des problèmes plus généraux. En effet, on a remarqué qu'une habile désintégration permet souvent de se ramener à la dimension un, ce qui permet d'utiliser les méthodes afférentes pour obtenir un premier résultat, que l'on fait ensuite évoluer pour gagner en précision.Je montre ici l'efficacité de cette approche, en revenant sur deux problèmes déjà résolus partiellement de cette manière, et en complétant la réponse qui en avait été donnée.Le premier problème concerne le calcul de l'application de Yann Brenier. En effet, Guillaume Carlier, Alfred Galichon et Filippo Santambrogio ont prouvé que celle-ci peut être obtenue grâce à une équation différentielle, pour laquelle une condition initiale est donnée par le réarrangement de Knothe-Rosenblatt (lui-même défini via une succession de transformations unidimensionnelles). Ils n'ont cependant traité que des mesures finales discrètes ; j'étends leur résultat aux cas continus. L'équation de Monge-Ampère, une fois dérivée, donne une EDP pour le potentiel de Kantorovitch; mais pour obtenir une condition initiale, il faut utiliser le théorème des fonctions implicites de Nash-Moser.Le chapitre 1 rappelle quelques résultats essentiels de la théorie du transport optimal, et le chapitre 2 est consacré au théorème de Nash-Moser. J'expose ensuite mes propres résultats dans le chapitre 3, et leur implémentation numérique dans le chapitre 4.Enfin, le dernier chapitre est consacré à l'algorithme IDT, développé par François Pitié, Anil C. Kokaram et Rozenn Dahyot. Celui-ci construit une application de transport suffisamment proche de celle de M. Brenier pour convenir à la plupart des applications. Une interprétation en est proposée en termes de flot de gradients dans l'espace des probabilités, avec pour fonctionnelle la distance de Wasserstein projetée. Je démontre aussi l'équivalence de celle-ci avec la distance usuelle de Wasserstein.In dimension one, optimal transportation is rather straightforward. The easiness with which a solution can be obtained in that setting has recently been used to tackle more general situations, each time thanks to the same method. First, disintegrate your problem to go back to the unidimensional case, and apply the available 1D methods to get a first result; then, improve it gradually using some evolution process.This dissertation explores that direction more thoroughly. Looking back at two problems only partially solved this way, I show how this viewpoint in fact allows to go even further.The first of these two problems concerns the computation of Yann Brenier's optimal map. Guillaume Carlier, Alfred Galichon, and Filippo Santambrogio found a new way to obtain it, thanks to an differential equation for which an initial condition is given by the Knothe-Rosenblatt rearrangement. (The latter is precisely defined by a series of unidimensional transformations.) However, they only dealt with discrete target measures; I~generalize their approach to a continuous setting. By differentiation, the Monge-Ampère equation readily gives a PDE satisfied by the Kantorovich potential; but to get a proper initial condition, it is necessary to use the Nash-Moser version of the implicit function theorem.The basics of optimal transport are recalled in the first chapter, and the Nash-Moser theory is exposed in chapter 2. My results are presented in chapter 3, and numerical experiments in chapter 4.The last chapter deals with the IDT algorithm, devised by François Pitié, Anil C. Kokaram, and Rozenn Dahyot. It builds a transport map that seems close enough to the optimal map for most applications. A complete mathematical understanding of the procedure is, however, still lacking. An interpretation as a gradient flow in the space of probability measures is proposed, with the sliced Wasserstein distance as the functional. I also prove the equivalence between the sliced and usual Wasserstein distances.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF
Proceedings of minisemester on evolution of interfaces, Sapporo 2010
conf: Special Project A, Proceedings of minisemester on evolution of interfaces, Sapporo (Department of Mathematics, Hokkaido University, July 12- August 13, 2010
It\^o calculus and jump diffusions for -L\'evy processes
The paper considers the integration theory for -L\'evy processes with
finite activity. We introduce the It\^o-L\'evy integrals, give the It\^o
formula for them and establish SDE's, BSDE's and decoupled FBSDE's driven by
-L\'evy processes. In order to develop such a theory, we prove two key
results: the representation of the sublinear expectation associated with a
-L\'evy process and a characterization of random variables in
in terms of their quasi-continuity