Anya
Katsevich
MIT, Cambridge, MA
Research Seminar on Mathematical Statistics  📅
Homepage
Head
Alexandra Carpentier,
Sonja Greven,
Wolfgang Karl Härdle,
Markus ReiĂź, and
Vladimir Spokoiny
Usual time
Wednesdays from 10:00 to 12:00
Usual venue
Weierstrass-Institut fĂĽr Angewandte Analysis und Stochastik
Erhard-Schmidt-Raum
Mohrenstrasse 39
10117 Berlin
Number of talks
24
July 2024
Celine
Duval
Université de Lille
June 2024
Clément
Berenfeld
Universität Potsdam
A theory of stratification learning
Abstract.
Given i.i.d. sample from a stratified mixture of immersed manifolds of different dimensions, we study the minimax estimation of the underlying stratified structure. We provide a constructive algorithm allowing to estimate each mixture component at its optimal dimension-specific rate adaptively. The method is based on an ascending hierarchical co-detection of points belonging to different layers, which also identifies the number of layers and their dimensions, assigns each data point to a layer accurately, and estimates tangent spaces optimally. These results hold regardless of any ambient assumption on the manifolds or on their intersection configurations. They open the way to a broad clustering framework, where each mixture component models a cluster emanating from a specific nonlinear correlation phenomenon.
Marc
Hallin
Université Libre de Bruxelles
The long quest for quantiles and ranks in Rd and on manifolds
Abstract.
Quantiles are a fundamental concept in probability, and an essential tool in statistics, from descriptive to inferential. Still, despite half a century of attempts, no satisfactory and fully agreed-upon definition of the concept, and the dual notion of ranks, is available beyond the well-understood case of univariate variables and distributions. The need for such a definition is particularly critical for varia- bles taking values in Rd, for directional variables (values on the hypersphere), and, more generally, for variables with values on manifolds. Unlike the real line, indeed, no canonical ordering is available on the- se domains. We show how measure transportation brings a solution to this problem by characterizing distribution-specific (data-driven, in the empirical case) orderings and center-outward distribution and quantile functions (ranks and signs in the empirical case) that satisfy all the properties expected from such concepts while reducing, in the case of real-valued variables, to the classical univariate notion.
Jia-Jie
Zhu
WIAS Berlin
Wasserstein and beyond: Optimal transport and gradient flows for machine learning and optimization
Abstract.
In the first part of the talk, I will provide an overview of gradient flows over non-negative and probability measures and their application in modern machine learning tasks, such as variational inference, sampling, training of over-parameterized models, and robust optimization. Then, I will present our recent results on the analysis of a couple of particularly relevant gradient flows, including the settings of Wasserstein, Hellinger/Fisher-Rao, and reproducing kernel Hilbert space. The focus is on the global exponential decay of the entropy functionals along the gradient flows such as Hellinger-Kantorovich (a.k.a. Wasserstein-Fisher-Rao) and a new type of gradient flow geometries that guarantee convergence of minimizing a maximum-mean discrepancy, which we term the interaction-force transport.
May 2024
Tailen
Hsing
University of Michigan
A functional-data perspective in spatial data analysis
Abstract.
More and more spatiotemporal data nowadays can be viewed as functional data. The first part of the talk focuses on the Argo data, which is a modern oceanography dataset that provides unprecedented global coverage of temperature and salinity measurements in the upper 2,000 meters of depth of the ocean. I will discuss a functional kriging approach to predict temperature and salinity as a smooth function of depth, as well as a co-kriging approach of predicting oxygen concentration based on temperature and salinity data. In the second part of the talk, I will give an overview on some related topics, including spectral density estimation and variable selection for functional data.
Vladimir
Spokoiny
WIAS Berlin
Gaussian variational inference in high dimension
Abstract.
We consider the problem of approximating a high-dimensional distribution by a Gaussian one by minimizing the Kullback-Leibler divergence. The main result extends Katsevich and Rigollet (2023) and claims that the minimiser can be well approximated by the Gaussian distribution with the mean and variance as for the underlying measure. We also describe the accuracy of approximation and the range of applicability for such approximation in terms of efficient dimension. The obtained results can be used for analysis of various sampling scheme in optimization.
Fabian
Telschow
HU Berlin
Estimation of the expected Euler characteristic of excursion sets of random fields and applications to simultaneous confidence bands
Abstract.
The expected Euler characteristic (EEC) of excursion sets of a smooth Gaussian-related random field over a compact manifold can be used to approximate the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC of a Gaussian-related field is expressed by the Gaussian kinematic formula (GKF) as a finite sum of known functions multiplied by the Lipschitz–Killing curvatures (LKCs) of the generating Gaussian field. In the first part of this talk we present consistent estimators of the LKCs as linear projections of ''pinned" Euler characteristic (EC) curves obtained from realizations of zero-mean, unit variance Gaussian processes. As observed data seldom is Gaussian, we generalize these LKC estimators by an unusual use of the Gaussian multiplier bootstrap to obtain consistent estimates of the LKCs of Gaussian limiting fields of non-stationary statistics. In the second part, we explain applications of LKC estimation and the GKF to simultaneous familywise error rate inference, for example, by constructing simultaneous confidence bands and CoPE sets for spatial functional data over complex domains such as fMRI and climate data and discuss their benefits and drawbacks compared to other methodologies.
Georg
Keilbar
HU Berlin
and
Ratmir
Miftachov
HU Berlin
April 2024
Nicolas
Verzelen
INRAE Montpellier
Computational trade-offs in high-dimensional clustering
Gil
Kur
ETH ZĂĽrich
Connections between minimum norm interpolation and local theory of Banach spaces
February 2024
Martin
Wahl
Universität Bielefeld
Heat kernel PCA with applications to Laplacian eigenmaps
Abstract.
Laplacian eigenmaps and diffusion maps are nonlinear dimensionality reduction methods that use the eigenvalues and eigenvectors of (un)normalized graph Laplacians. Both methods are applied when the data is sampled from a low-dimensional manifold, embedded in a high-dimensional Euclidean space. From a mathematical perspective, the main problem is to understand these empirical Laplacians as spectral approximations of the underlying Laplace-Beltrami operator. In this talk, we study Laplacian eigenmaps through the lens of kernel PCA, and consider the heat kernel as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.
Evgenii
Chzhen
LMO Orsay, Paris
January 2024
Gianluca
Finocchio
Universität Wien
An extended latent factor framework for ill-posed linear regression
Abstract.
The classical latent factor model for linear regression is extended by assuming that, up to an unknown orthogonal transformation, the features consist of subsets that are relevant and irrelevant for the response. Furthermore, a joint low-dimensionality is imposed only on the relevant features vector and the response variable. This framework allows for a comprehensive study of the partial-least-squares (PLS) algorithm under random design. In particular, a novel perturbation bound for PLS solutions is proven and the high-probability L²-estimation rate for the PLS estimator is obtained. This novel framework also sheds light on the performance of other regularisation methods for ill-posed linear regression that exploit sparsity or unsupervised projection. The theoretical findings are confirmed by numerical studies on both real and simulated data.
Simon
Wood
University of Edinburgh
On neighbourhood cross validation
Abstract.
Cross validation comes in many varieties, but some of the more interesting flavours require multiple model fits with consequently high cost. This talk shows how the high cost can be side-stepped for a wide range of models estimated using a quadratically penalized smooth loss, with rather low approximation error. Once the computational cost has the same leading order as a single model fit, it becomes feasible to efficiently optimize the chosen cross-validation criterion with respect to multiple smoothing/precision parameters. Interesting applications include cross-validating smooth additive quantile regression models, and the use of leave-out-neighbourhood cross validation for dealing with nuisance short range autocorrelation. The link between cross validation and the jackknife can be exploited to obtain reasonably well calibrated uncertainty quantification in these cases
Matteo
Giordano
UniversitĂ degli Studi di Torino
Likelihood methods for low frequency diffusion data
Abstract.
The talk will consider the problem of nonparametric inference in multi-dimensional diffusion models from low-frequency data. Implementation of likelihood-based procedures in such settings is a notoriously delicate task, due to the computational intractability of the likelihood. For the nonlinear inverse problem of inferring the diffusivity in a stochastic differential equation, we propose to exploit the underlying PDE characterisation of the transition densities, which allows the numerical evaluation of the likelihood via standard numerical methods for elliptic eigenvalue problems. A simple Metropolis-Hastings-type MCMC algorithm for Bayesian inference on the diffusivity is then constructed, based on Gaussian process priors. Furthermore, the PDE approach also yields a convenient characterisation of the gradient of the likelihood via perturbation techniques for parabolic PDEs, allowing the construction of gradient-based inference methods including MLE and Langevin-type MCMC. The performance of the algorithms is illustrated via the results of numerical experiments. Joint work with Sven Wang.
Eric
Moulines
Ecole Polytechnique
Score-based diffusion models and applications
Abstract.
Deep generative models represent an advanced frontier in machine learning. These models are adept at fitting complex data sets, whether they consist of images, text or other forms of high-dimensional data. What makes them particularly noteworthy is their ability to provide independent samples from these complicated distributions at a cost that is both computationally efficient and resource efficient. However, the task of accurately sampling a target distribution presents significant challenges. These challenges often arise from the high dimensionality, multimodality or a combination of these factors. This complexity can compromise the effectiveness of traditional sampling methods and make the process either computationally prohibitive or less accurate. In my talk, I will address recent efforts in this area that aim to improve traditional inference and sampling algorithms. My major focus will be on score-based diffusion models. By utilizing the concept of score matching and time-reversal of stochastic differential equations, they offer a novel and powerful approach to generating high-quality samples. I will discuss how these models work, their underlying principles and how they are used to overcome the limitations of conventional methods. The talk will also cover practical applications, demonstrating their versatility and effectiveness in solving complex real-world problems.
December 2023
Laura
Sangalli
MOX Milano, Italien
Physics-informed spatial and functional data analysis
Abstract.
Recent years have seen an explosive growth in the recording of increasingly complex and high-dimensional data, whose analysis calls for the definition of new methods, merging ideas and approaches from statistics and applied mathematics. My talk will focus on spatial and functional data observed over non-Euclidean domains, such as linear networks, two-dimensional manifolds and non-convex volumes. I will present an innovative class of methods, based on regularizing terms involving Partial Differential Equations (PDEs), defined over the complex domains being considered. These Physics-Informed statistical learning methods enable the inclusion of the available problem specific information, suitably encoded in the regularizing PDE. Illustrative applications from environmental and life sciences will be presented.
Boris
Buchmann
ANU Canberra, Australia
Weak subordination of multivariate Levy processes
Abstract.
Subordination is the operation which evaluates a Levy process at a subordinator, giving rise to a pathwise construction of a "time-changed" process. In probability semigroups, subordination was applied to create the variance gamma process, which is prominently used in financial modelling. However, subordination may not produce a levy process unless the subordinate has independent components or the subordinate has indistinguishable components. We introduce a new operation known as weak subordination that always produces a Levy process by assigning the distribution of the subordinate conditional on the value of the subordinator, and matches traditional subordination in law in the cases above. Weak subordination is applied to extend the class of variance-generalised gamma convolutions and to construct the weak variance-alpha-gamma process. The latter process exhibits a wider range of dependence than using traditional subordination. Joint work with Kevin W. LU - Australian National University (Australia) & Dilip B. Madan - University of Maryland (USA)
November 2023
Martin
Spindler
Universität Hamburg
High-dimensional L2-boosting: Rate of convergence (hybrid talk)
Marc
Hoffmann
Université Paris-Dauphine
On estimating multidimensional diffusions from discrete data
Sven
Wang
Humboldt-Universität zu Berlin
Victor
Panaretos
EPFL Lausanne
Optimal transport for covariance operators
Abstract.
Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on covariance operators, and the intrinsic infinite-dimensionality and of these operators. I will describe how the geometry and tools of optimal transportation can be leveraged to construct natural and effective statistical summaries and inference tools for covariance operators, taking full advantage of the nature of their ambient space. Based on joint work with Valentina Masarotto (Leiden), Leonardo Santoro (EPFL), and Yoav Zemel (EPFL).
October 2023
Denis
Belomestny
Universität Duisburg-Essen
Provable benefits of policy learning from human preferences