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18 results on '"Department of Mathematics [Univ California San Diego] (MATH - UC San Diego)"'

1. On the Estimation of Latent Distances Using Graph Distances

Author

Bruno Pelletier, Ery Arias-Castro, Antoine Channarond, Nicolas Verzelen, Department of Mathematics [San Diego], University of California [San Diego] (UC San Diego), University of California-University of California, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), National Science Foundation (NSF): DMS 1513465, DMS 1916071, French Region Normandie : RIN 17B01101GR, ANR-18-CE40-0014,SMILES,Modélisation et Inférence Statistique pour l'Apprentissage non-supervisé à partir de Données Massives(2018), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Institut National de la Recherche Agronomique (INRA), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California (UC)-University of California (UC), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Agro - Montpellier SupAgro

Subjects
Statistics and Probability, multidimensional scaling, Latent positions, Graph embedding, Mathematics - Statistics Theory, Statistics Theory (math.ST), 02 engineering and technology, 01 natural sciences, Upper and lower bounds, Task (project management), Combinatorics, 010104 statistics & probability, Spatial network, Mathematics - Metric Geometry, Indicator function, graph distances, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], FOS: Mathematics, 0202 electrical engineering, electronic engineering, information engineering, Adjacency matrix, Multidimensional scaling, 0101 mathematics, [MATH]Mathematics [math], Mathematics, graph embedding, Metric Geometry (math.MG), 020206 networking & telecommunications, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], random geometric graphs, Graph (abstract data type), Statistics, Probability and Uncertainty
Abstract

International audience; We are given the adjacency matrix of a geometric graph and the task of recovering the latent positions. We study one of the most popular approaches which consists in using the graph distances and derive error bounds under various assumptions on the link function. In the simplest case where the link function is an indicator function, the bound is (nearly) optimal as it (nearly) matches an information lower bound.

Published
2021

2. Perturbation Bounds for Procrustes, Classical Scaling, and Trilateration, with Applications to Manifold Learning

Author

Arias-Castro, Ery, Javanmard, Adel, Pelletier, Bruno, Department of Mathematics [San Diego], University of California [San Diego] (UC San Diego), University of California-University of California, Department of Electrical Engineering [Tehran] (EE department of SUT [Tehran]), Sharif University of Technology [Tehran] (SUT), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California (UC)-University of California (UC), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
FOS: Computer and information sciences, Computer Science - Machine Learning, [INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG], [STAT.ML]Statistics [stat]/Machine Learning [stat.ML], Statistics - Machine Learning, FOS: Mathematics, Machine Learning (stat.ML), Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Machine Learning (cs.LG)
Abstract

One of the common tasks in unsupervised learning is dimensionality reduction, where the goal is to find meaningful low-dimensional structures hidden in high-dimensional data. Sometimes referred to as manifold learning, this problem is closely related to the problem of localization, which aims at embedding a weighted graph into a low-dimensional Euclidean space. Several methods have been proposed for localization, and also manifold learning. Nonetheless, the robustness property of most of them is little understood. In this paper, we obtain perturbation bounds for classical scaling and trilateration, which are then applied to derive performance bounds for Isomap, Landmark Isomap, and Maximum Variance Unfolding. A new perturbation bound for procrustes analysis plays a key role., Comment: 33 pages, 6 Figures

Published
2020

3. Functorial properties of generalised Steinberg representations

Author

Tobias Schmidt, Claus Sorensen, Julien Hauseux, King‘s College London, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Mathematics [San Diego], University of California [San Diego] ( UC San Diego ), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), University of California [San Diego] (UC San Diego), University of California-University of California, EP/L025302/1, Engineering and Physical Sciences Research Council, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), and University of California (UC)-University of California (UC)

Subjects
Reductive groups, 010103 numerical & computational mathematics, Commutative ring, 01 natural sciences, Extensions, Combinatorics, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], FOS: Mathematics, Number Theory (math.NT), Representation Theory (math.RT), 0101 mathematics, Steinberg representation, generalised Steinberg representations, Mathematics, Deformations, 22E50, 11F70, Algebra and Number Theory, Functor, Mathematics - Number Theory, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 010102 general mathematics, Reductive group, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Local fields, [ MATH.MATH-RT ] Mathematics [math]/Representation Theory [math.RT], 22E50, Generalised Steinberg representations, Mathematics - Representation Theory
Abstract

Let $G$ be the $F$-points of a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$ and $R$ be a commutative ring. Let $P=LU$ be a parabolic subgroup of $G$ and $Q$ be a parabolic subgroup of $G$ containing $P$. We study the functor $\mathrm{St}_Q^G$ taking a smooth $R$-representation $\sigma$ of $L$ which extends to a representation $\mathrm{e}_G(\sigma)$ of $G$ trivial on $U$ to the smooth $R$-representation $\mathrm{e}_G(\sigma) \otimes_R \mathrm{St}_Q^G(R)$ of $G$ where $\mathrm{St}_Q^G(R)$ is the generalised Steinberg representation., Comment: 14 pages, minor changes, final version

Published
2019

4. USEFUL BOUNDS ON THE EXTREME EIGENVALUES AND VECTORS OF MATRICES FOR HARPER'S OPERATORS

Author

Bump, Daniel, Diaconis, Persi, Hicks, Angela, Miclo, Laurent, Widom, Harold, Department of Statistics [Stanford], Stanford University, Department of Mathematics [San Diego], University of California [San Diego] (UC San Diego), University of California-University of California, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California (UC)-University of California (UC), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Abstract

International audience; In analyzing a simple random walk on the Heisenberg group we encounter the problem of bounding the extreme eigenvalues of an n × n matrix of the form M = C + D where C is a circulant and D a diagonal matrix. The discrete Schrödinger operators are an interesting special case. The Weyl and Horn bounds are not useful here. This paper develops three different approaches to getting good bounds. The first uses the geometry of the eigenspaces of C and D, applying a discrete version of the uncertainty principle. The second shows that, in a useful limit, the matrix M tends to the harmonic oscillator on L 2 (R) and the known eigenstructure can be transferred back. The third approach is purely probabilistic, extending M to an absorbing Markov chain and using hitting time arguments to bound the Dirichlet eigenvalues. The approaches allow generalization to other walks on other groups.

Published
2017

5. Deformation rings and parabolic induction

Author

Tobias Schmidt, Claus Sorensen, Julien Hauseux, King‘s College London, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Mathematics [San Diego], University of California [San Diego] ( UC San Diego ), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), University of California [San Diego] (UC San Diego), University of California-University of California, EP/L025302/1, Engineering and Physical Sciences Research Council, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), and University of California (UC)-University of California (UC)

Subjects
Pure mathematics, m-adically continuous representations, Bar (music), Lift (mathematics), FOS: Mathematics, Number Theory (math.NT), Representation Theory (math.RT), Mathematics, 22E50, 11F70, Algebra and Number Theory, Deformation (mechanics), Mathematics - Number Theory, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], deformations, smooth representations, Sigma, parabolic induction, Reductive group, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], 22E50, Parabolic induction, Dual polyhedron, Isomorphism, p-adic reductive groups, Mathematics - Representation Theory
Abstract

We study deformations of smooth mod $p$ representations (and their duals) of a $p$-adic reductive group $G$. Under some mild genericity condition, we prove that parabolic induction with respect to a parabolic subgroup $P=LN$ defines an isomorphism between the universal deformation rings of a supersingular representation $\bar{\sigma}$ of $L$ and of its parabolic induction $\bar{\pi}$. As a consequence, we show that every Banach lift of $\bar{\pi}$ is induced from a unique Banach lift of $\bar{\sigma}$., Comment: 28 pages, minor changes, final version

Published
2016

6. Remember the Curse of Dimensionality: The Case of Goodness-of-Fit Testing in Arbitrary Dimension

Author

Ery Arias-Castro, Venkatesh Saligrama, Bruno Pelletier, Department of Mathematics [San Diego], University of California [San Diego] ( UC San Diego ), Laboratoire de statistiques, Université Rennes 2 ( LABORATOIRE DE STATISTIQUES ), Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Electrical and Computer Engineering [Boston University] ( ECE ), Boston University [Boston] ( BU ), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California [San Diego] (UC San Diego), University of California (UC)-University of California (UC), Laboratoire de statistiques, Université Rennes 2 (LABORATOIRE DE STATISTIQUES), Université de Rennes 2 (UR2), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Department of Electrical and Computer Engineering [Boston University] (ECE), Boston University [Boston] (BU), University of California-University of California, Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Statistics and Probability, Statistics::Theory, 010102 general mathematics, Nonparametric statistics, Mathematics - Statistics Theory, Statistics Theory (math.ST), 01 natural sciences, 010104 statistics & probability, Dimension (vector space), Goodness of fit, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Econometrics, FOS: Mathematics, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], 0101 mathematics, Statistics, Probability and Uncertainty, Statistical theory, Mathematics, Curse of dimensionality
Abstract

Despite a substantial literature on nonparametric two-sample goodness-of-fit testing in arbitrary dimensions spanning decades, there is no mention there of any curse of dimensionality. Only more recently Ramdas et al. (2015) have discussed this issue in the context of kernel methods by showing that their performance degrades with the dimension even when the underlying distributions are isotropic Gaussians. We take a minimax perspective and follow in the footsteps of Ingster (1987) to derive the minimax rate in arbitrary dimension when the discrepancy is measured in the L2 metric. That rate is revealed to be nonparametric and exhibit a prototypical curse of dimensionality. We further extend Ingster's work to show that the chi-squared test achieves the minimax rate. Moreover, we show that the test can be made to work when the distributions have support of low intrinsic dimension. Finally, inspired by Ingster (2000), we consider a multiscale version of the chi-square test which can adapt to unknown smoothness and/or unknown intrinsic dimensionality without much loss in power., Comment: This version comes after the publication of the paper in the Journal of Nonparametric Statistics. The main change is to cite the work of Ramdas et al. Some very minor typos were also corrected

Published
2016

7. On the Estimation of the Gradient Lines of a Density and the Consistency of the Mean-Shift Algorithm

Author

Arias-Castro , Ery, Pelletier , Bruno, Mason , David, Department of Mathematics [San Diego], University of California [San Diego] ( UC San Diego ), Laboratoire de statistiques, Université Rennes 2 ( LABORATOIRE DE STATISTIQUES ), Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Applied Economics and Statistics, University of Delaware [Newark], Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California [San Diego] (UC San Diego), University of California (UC)-University of California (UC), Laboratoire de statistiques, Université Rennes 2 (LABORATOIRE DE STATISTIQUES), Université de Rennes 2 (UR2), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), University of California-University of California, Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
nonparametric clustering, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], density estimation, gradient lines, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], mean-shift
Abstract

International audience; We consider the problem of estimating the gradient lines of a density, which can be used to cluster points sampled from that density, for example via the mean-shift algorithm of Fukunaga and Hostetler (1975). We prove general convergence bounds that we then specialize to kernel density estimation.

Published
2016

8. Critical branching Brownian motion with absorption: Particle configurations

Author

Julien Berestycki, Jason Schweinsberg, Nathanaël Berestycki, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California [San Diego] (UC San Diego), University of California (UC)-University of California (UC), Department of Mathematics at the University of California at San Diego, and University of California-University of California

Subjects
Statistics and Probability, Particle number, Statistics & Probability, 60J65, 60J80, 60J25, Critical phenomena, Branching (polymer chemistry), math.PR, 01 natural sciences, 010104 statistics & probability, 60J25, FOS: Mathematics, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Brownian motion, Mathematics, 60J80, Statistics, Probability (math.PR), 010102 general mathematics, Mathematical analysis, Primary 60J65, critical phenomena, Yaglom limit laws, Brownian excursion, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Extinction time, Branching Brownian motion, Primary 60J65, Secondary 60J80, 60J25, Statistics, Probability and Uncertainty, Mathematics - Probability, Secondary 60J80
Abstract

International audience; We consider critical branching Brownian motion with absorption, in which there is initially a single particle at x > 0, particles move according to independent one-dimensional Brownian motions with the critical drift of -root 2, and particles are absorbed when they reach zero. Here we obtain asymptotic results concerning the behavior of the process before the extinction time, as the position x of the initial particle tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle. We also obtain asymptotic results for the configuration of particles at a typical time.

Published
2015

9. An Exercise (?) in Fourier Analysis on the Heisenberg Group

Author

Laurent Miclo, Harold Widom, Angela Hicks, Persi Diaconis, Daniel Bump, Department of Statistics [Stanford], Stanford University, Department of Mathematics [San Diego], University of California [San Diego] (UC San Diego), University of California-University of California, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California (UC)-University of California (UC), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, Uniform distribution (continuous), Group (mathematics), 60J10, 60B15, Probability (math.PR), 010102 general mathematics, Spectrum (functional analysis), General Medicine, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], symbols.namesake, Bounding overwatch, Fourier analysis, 0103 physical sciences, FOS: Mathematics, Heisenberg group, symbols, Order (group theory), 010307 mathematical physics, 0101 mathematics, Variety (universal algebra), Mathematics - Probability, Mathematics
Abstract

Let H(n) be the group of 3x3 uni-uppertriangular matrices with entries in Z/nZ, the integers mod n. We show that the simple random walk converges to the uniform distribution in order n^2 steps. The argument uses Fourier analysis and is surprisingly challenging. It introduces novel techniques for bounding the spectrum which are useful for a variety of walks on a variety of groups., 24 pages, 6 figures

Published
2015

10. On the Consistency of the Crossmatch Test

Author

Ery Arias-Castro, Bruno Pelletier, Department of Mathematics [San Diego], University of California [San Diego] ( UC San Diego ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California [San Diego] (UC San Diego), University of California (UC)-University of California (UC), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Université de Rennes 2 (UR2), N00014-13-1-0257, Office of Naval Research, University of California-University of California, AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), and Université de Rennes (UNIV-RENNES)

Subjects
Statistics and Probability, consistency, Goodness-of-fit testing in arbitrary dimensions, Applied Mathematics, Process (computing), Mathematics - Statistics Theory, 0102 computer and information sciences, Statistics Theory (math.ST), 16. Peace & justice, 01 natural sciences, Test (assessment), the crossmatch test, 010104 statistics & probability, 010201 computation theory & mathematics, Consistency (statistics), [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Calculus, FOS: Mathematics, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], 0101 mathematics, Statistics, Probability and Uncertainty, Graph-based tests, Mathematics
Abstract

International audience; Rosenbaum (2005) proposed the crossmatch test for two-sample goodness-of-fit testing in arbitrary dimensions. We prove that the test is consistent against all fixed alternatives. In the process, we develop a general consistency result based on (Henze & Penrose, 1999) that applies more generally.

Published
2015
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11. Critical branching Brownian motion with absorption: survival probability

Author

Julien Berestycki, Nathanaël Berestycki, Jason Schweinsberg, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Mathematics at the University of California at San Diego, University of California [San Diego] (UC San Diego), University of California-University of California, Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), and University of California (UC)-University of California (UC)

Subjects
Statistics and Probability, Extinction time, 60J65, 60J80, 60J25, Statistics & Probability, 01 natural sciences, Upper and lower bounds, math.PR, Critical phenomena, 010104 statistics & probability, Mathematics::Probability, 60J25, FOS: Mathematics, Calculus, Almost surely, Statistical physics, 0101 mathematics, Brownian motion, ComputingMilieux_MISCELLANEOUS, Mathematics, Geometric Brownian motion, Fractional Brownian motion, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Statistics, Primary 60J65, Brownian excursion, Pure Mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Branching processes, Reflected Brownian motion, Diffusion process, Branching Brownian motion, Primary 60J65, Secondary 60J80, 60J25, Survival probability, Statistics, Probability and Uncertainty, Mathematics - Probability, Analysis, Secondary 60J80
Abstract

We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of $$-\sqrt{2}$$ . Kesten (Stoch Process 7:9–47, 1978) showed that almost surely this process eventually dies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time $$t$$ . These bounds improve upon results of Kesten (Stoch Process 7:9–47, 1978), and partially confirm nonrigorous predictions of Derrida and Simon (EPL 78:60006, 2007).

Published
2014

12. The affine invariant of generalized semitoric systems

Author

Pelayo, ��lvaro, Ratiu, Tudor S., Ngoc, San V��, Department of Mathematics [San Diego], University of California [San Diego] (UC San Diego), University of California-University of California, Département de Mathématiques - EPFL, Ecole Polytechnique Fédérale de Lausanne (EPFL), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 1518420, Division of Mathematical Sciences, ANR NOSEVOL grant, 200021-140238, Swiss NSF, NCCR SwissMAP, 11.G34.31.0054, Russian Federation, Lomonosov Moscow State University, University of California [San Diego] ( UC San Diego ), Ecole Polytechnique Fédérale de Lausanne ( EPFL ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California (UC)-University of California (UC), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
[ MATH.MATH-GT ] Mathematics [math]/Geometric Topology [math.GT], Mathematics - Geometric Topology, geometric topology, Mathematics - Symplectic Geometry, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], FOS: Mathematics, Symplectic Geometry (math.SG), Geometric Topology (math.GT), [ MATH.MATH-SG ] Mathematics [math]/Symplectic Geometry [math.SG], 37J35, 70H06, 53D20, 57M60, 57R17, 57S25, 58D27, sympletic geometry, Mathematics::Symplectic Geometry, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG]
Abstract

A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not necessarily proper. These systems can exhibit focus-focus singularities, which correspond to fibers of F which are topologically multipinched tori. The image F(M) is a singular affine manifold which contains a distinguished set of isolated points in its interior: the focus-focus values {(x_i,y_i)} of F. By performing a vertical cutting procedure along the lines {x:=x_i}, we construct a homeomorphism f : F(M) --> f(F(M)), which restricts to an affine diffeomorphism away from these vertical lines, and generalizes a construction of Vu Ngoc. The set \Delta:=f(F(M)) in R^2 is a symplectic invariant of (M,\omega,F), which encodes the affine structure of F. Moreover, \Delta may be described as a countable union of planar regions of four distinct types, where each type is defined as the region bounded between the graphs of two functions with various properties (piecewise linear, continuous, convex, etc). If F is a toric system, \Delta is a convex polygon (as proven by Atiyah and Guillemin-Sternberg) and f is the identity., Comment: 29 pages, 15 figures

Published
2013

13. On the convergence of maximum variance unfolding

Author

Arias-Castro , Ery, Pelletier , Bruno, Department of Mathematics [San Diego], University of California [San Diego] (UC San Diego), University of California-University of California, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), University of California [San Diego] ( UC San Diego ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California (UC)-University of California (UC), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
FOS: Computer and information sciences, Maximum Variance Unfolding, Isometric embedding, U-processes, empirical processes, proximity graphs, [STAT.ML]Statistics [stat]/Machine Learning [stat.ML], Statistics - Machine Learning, Machine Learning (stat.ML), 62G05, 62G20, [ STAT.ML ] Statistics [stat]/Machine Learning [stat.ML]
Abstract

International audience; Maximum Variance Unfolding is one of the main methods for (nonlinear) dimensionality reduction. We study its large sample limit, providing specific rates of convergence under standard assumptions. We find that it is consistent when the underlying submanifold is isometric to a convex subset, and we provide some simple examples where it fails to be consistent.

Published
2013

14. The Normalized Graph Cut and Cheeger Constant: from Discrete to Continuous

Author

Pierre Pudlo, Ery Arias-Castro, Bruno Pelletier, Department of Mathematics [San Diego], University of California [San Diego] ( UC San Diego ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Grant from the US National Science Foundation (DMS-06-03890), Clustering in High Dimension: Algorithms and Applications, CLARA, ANR-09-BLAN-0051-01,Clustering in High Dimension: Algorithms and Applications, CLARA, ANR-09-BLAN-0051-01, University of California [San Diego] (UC San Diego), University of California-University of California, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), ANR-09-BLAN-0051,CLARA,Classification non-supervisée en grande dimension : algorithmes et applications(2009), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California (UC)-University of California (UC), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Normalization (statistics), U-process, Mathematics - Statistics Theory, Statistics Theory (math.ST), 01 natural sciences, empirical process, Combinatorics, 010104 statistics & probability, conductance of a graph, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Cut, Subsequence, FOS: Mathematics, 62G05, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], 0101 mathematics, 62G20, Mathematics, spectral clustering, empirical processes, Euclidean space, Applied Mathematics, 010102 general mathematics, Conductance, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH], Spectral clustering, U-processes, Sample size determination, Bounded function, MSC 62G05 62G20, neighborhood graph, Cheeger isoperimetric constant of a manifold
Abstract

LetMbe a bounded domain ofwith a smooth boundary. We relate the Cheeger constant ofMand the conductance of a neighborhood graph defined on a random sample fromM. By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set ofM.

Published
2012

15. Survival of near-critical branching Brownian motion

Author

Jason Schweinsberg, Nathanaël Berestycki, Julien Berestycki, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Mathematics at the University of California at San Diego, University of California [San Diego] (UC San Diego), University of California-University of California, ANR-08-BLAN-0220,MADCOF,Méthodes Aléatoires et Déterministes pour les processus de COllision, coalescence, de Fragmentation(2008), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), and University of California (UC)-University of California (UC)

Subjects
Near critical, Probability (math.PR), 010102 general mathematics, Zero (complex analysis), Statistical and Nonlinear Physics, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Survival probability, FOS: Mathematics, Traveling wave, 60J99 (Primary), 60J80, 60F17, 60G15 (Secondary), 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematical Physics, Brownian motion, Positive probability, Mathematics - Probability, Mathematics, Mathematical physics
Abstract

International audience; Consider a system of particles performing branching Brownian motion with negative drift mu = root 2-epsilon and killed upon hitting zero. Initially there is one particle at x > 0. Kesten (Stoch. Process. Appl. 7: 9-47, 1978) showed that the process survives with positive probability if and only if e > 0. Here we are interested in the asymptotics as e. 0 of the survival probability Q(mu)(x). It is proved that if L = p/v e then for all x. R, lim(epsilon -> 0) Q(mu)(L + x) = theta(x) is an element of (0, 1) exists and is a traveling wave solution of the Fisher-KPP equation. Furthermore, we obtain sharp asymptotics of the survival probability when x < L and L - x -> infinity. The proofs rely on probabilistic methods developed by the authors in (Berestycki et al. in arXiv:1001.2337, 2010). This completes earlier work by Harris, Harris and Kyprianou (Ann. Inst. Henri Poincare Probab. Stat. 42: 125-145, 2006) and confirms predictions made by Derrida and Simon (Europhys. Lett. 78: 60006, 2007), which were obtained using nonrigorous PDE methods.

Published
2010

16. Convergence de la constante de Cheeger de graphes de voisinage

Author

Arias-Castro , Ery, Pelletier , Bruno, Pudlo , Pierre, Department of Mathematics [San Diego], University of California [San Diego] (UC San Diego), University of California-University of California, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Université Montpellier 2 - Sciences et Techniques (UM2), University of California [San Diego] ( UC San Diego ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Université Montpellier 2 - Sciences et Techniques ( UM2 ), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California (UC)-University of California (UC), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH]
Abstract

International audience; Nous nous intéressons dans ce travail aux ensembles minimisant la constante de Cheeger d'un sous-ensemble $\mathcalM$ de $\mathbbR^d$. Cette dernière minimise le rapport d'un périmètre à un volume parmi tous les sous-ensembles de $\mathcalM$. Étant donné un $n$-échantillon issu de la mesure uniforme sur $\mathcalM$, nous introduisons une version régularisée de la conductance du graphe de voisinage construit sur l'échantillon. Nous établissons alors la convergence de la conductance régularisée vers la constante de Cheeger de $\mathcalM$. En outre, nous montrons la convergence des suites de partitions optimales du graphe vers les ensembles de Cheeger de $\mathcalM$ pour la topologie de $L^1(\mathcalM)$.

Published
2010

17. The genealogy of branching Brownian motion with absorption

Author

Julien Berestycki, Jason Schweinsberg, Nathanaël Berestycki, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California [San Diego] (UC San Diego), University of California (UC)-University of California (UC), ANR-06-BLAN-0113,MAEV,Modèles Aléatoires de l'Evolution du Vivant(2006), Department of Mathematics at the University of California at San Diego, and University of California-University of California

Subjects
Statistics and Probability, Scale (ratio), Population, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Bolthausen–Sznitman coalescent, FOS: Physical sciences, 01 natural sciences, continuous-state branching processes, Coalescent theory, Branching (linguistics), 010104 statistics & probability, Mathematics::Probability, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, Quantitative Biology::Populations and Evolution, 0101 mathematics, education, Mathematical Physics, Brownian motion, ComputingMilieux_MISCELLANEOUS, Branching process, Mathematics, 60J80, education.field_of_study, 010102 general mathematics, Probability (math.PR), Zero (complex analysis), Mathematical Physics (math-ph), 16. Peace & justice, Genealogy, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60F17, 60G15, Branching Brownian motion, 60J99, Statistics, Probability and Uncertainty, Constant (mathematics), Mathematics - Probability
Abstract

We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles. We show that the characteristic time scale for the evolution of this population is of order $(\log N)^3$, in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu's continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen-Sznitman coalescent. This validates the nonrigorous predictions by Brunet, Derrida, Muller and Munier for a closely related model., Comment: Published in at http://dx.doi.org/10.1214/11-AOP728 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2010
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18. Fiber connectivity and bifurcation diagrams of almost toric integrable systems

Author

San Vu Ngoc, Álvaro Pelayo, Tudor S. Ratiu, Department of Mathematics [San Diego], University of California [San Diego] (UC San Diego), University of California-University of California, Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Ecole Polytechnique Fédérale de Lausanne (EPFL), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California (UC)-University of California (UC), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Pure mathematics, Integrable system, 37J35, 53D05, 53D20, Lagrangian fibrations, Fibration, affine structures, 16. Peace & justice, Bifurcation diagram, Hamiltonian system, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], symplectic geometry, Integrable systems, quantum systems, Gravitational singularity, Geometry and Topology, [MATH]Mathematics [math], Mathematics::Symplectic Geometry, singularities, Bifurcation, Symplectic geometry, Mathematics, Symplectic manifold
Abstract

International audience; We describe the bifurcation diagrams of almost toric integrable Hamiltonian systems on a four dimensional symplectic manifold MM, not necessarily compact. We prove that, under a weak assumption, the connectivity of the fibers of the induced singular Lagrangian fibration M→R^2 can be detected from the bifurcation diagram alone. In this case, it is possible to give a detailed description of the image of the fibration.

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