Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Für den Bereich Statistik

A. Carpentier, S. Greven, W. Härdle, M. Reiß, V. Spokoiny

Ort

Weierstrass-Institut für Angewandte Analysis und Stochastik
HVP 11 a,  R.313 bitte beachten Sie die Raumänderung!!!
Mohrenstrasse 39
10117 Berlin

Zeit

mittwochs, 10.00 - 12.00 Uhr

Programm

 

16. Oktober 2024

Botond Szabo (Bocconi Milan)

Privacy constrained semiparametric inference

Abstract: For semi-parametric problems differential private estimators are typically constructed in a case-by-case basis. In this work we develop a privacy constrained semi-parametric plug-in approach, which can be used in general, over a collection of semi-parametric problems. We derive minimax lower and matching upper bounds for this approach and provide an adaptive procedure in case of irregular (atomic) functionals. Joint work with Lukas Steinberger (Vienna) and Thibault Randrianarisoa (Toronto, Vector Institute).

 

23. Oktober 2024

Weining Wang (University of Groningen)

Conditional Nonparametric Variable Screening Test via Neural Network Factor Regression

Abstract: We propose a conditional variable screening test for non-parametric regression.  To render our test effective when facing predictors with high or even diverging dimension, we assume that the observed predictors arise from a factor model where the factors are latent but low-dimensional.  Our test statistics are based on the estimated partial derivative of the regression function in the screening variable when conditioning on the extracted proxies for the factors.  Hence, our test reveals how much predictors contribute to non-parametric regression after accounting for the factors.  Our derivative estimator is the convolution of a deep neural network regression estimator and a smoothing kernel.  We demonstrate that when the neural network could scale up as the sample size grows, unlike estimating the regression function itself, it is important to smooth the partial derivative of the neural network estimator to recover the desired convergence rate for the derivative.  Moreover, our screening test achieves asymptotic normality under the null after finely centering our test statistics, as well as consistency for local alternatives under mild conditions.  We demonstrate the performance of our test in a simulation study and two real world applications.

30. Oktober 2024

Olga Klopp (ESSEC Business School, Paris) 

Adaptive density estimation under low-rank constraints

Abstract: In this talk, we address the challenge of bivariate probability density estimation under low-rank constraints for both discrete and continuous distributions. For discrete distributions, we model the target as a low-rank probability matrix. In the continuous case, we assume the density function is Lipschitz continuous over an unknown compact rectangular support and can be decomposed into a sum of K separable components, each represented as a product of two one-dimensional functions. We introduce an estimator that leverages these low-rank constraints, achieving significantly improved convergence rates. We also derive lower bounds for both discrete and continuous cases, demonstrating that our estimators achieve minimax optimal convergence rates within logarithmic factors.

06. November 2024

     Vladimir Spokoiny (WIAS und HUB)

       Regression estimation and inference in high dimension

Abstract: The talk discusses a general non-asymptotic and non-minimax approach to statistical     estimation and inference with applications to nonlinear regression. The main results provide finite-sample Fisher and Wilks expansions for the maximum-likelihood estimator with an explicit remainder in terms of the effective dimension of the problem.

13. November 2024 findet in Adlershof statt

Vladimir Spokoinys Geburtstagskolloquium (WIAS/ HU)

Prof. Spokoinys Geburtstagskolloquium anlässlich seines 65. Geburtstag

für weitere Information https://wias-berlin.de/workshops/Spokoiny2024/

20. November 2024

Bernhard Stankewitz (Universität Potsdam) 

Contraction rates for conjugate gradient and Lanczos approximate posteriors in Gaussian process regression

Abstract: Due to their flexibility and theoretical tractability Gaussian process (GP) regression models have become a central topic in modern statistics and machine learning. While the true posterior in these models is given explicitly, numerical evaluations depend on the inversion of the augmented kernel matrix K + σ^2 I, which requires up to O(n^{^3}) operations. For large sample sizes n, which are typically given in modern applications, this is computationally infeasible and necessitates the use of an approximate version of the posterior. Although such methods are widely used in practice, they typically have very limtied theoretical underpinning.

In this context, we analyze a class of recently proposed approximation algorithms from the field of Probabilistic numerics. They can be interpreted in terms of Lanczos approximate eigenvectors of the kernel matrix or a conjugate gradient approximation of the posterior mean, which are particularly advantageous in truly large scale applications, as they are fundamentally only based on matrix vector multiplications amenable to the GPU acceleration of modern software frameworks. We combine result from the numerical analysis literature with state of the art concentration results for spectra of kernel matrices to obtain minimax contraction rates. Our theoretical findings are illustrated by numerical experiments.

27. November 2024    

Julien Chhor (Toulouse School of Economics)

Locally sharp goodness-of-fit testing in sup norm for high-dimensional counts

Abstract: We consider testing the goodness-of-fit of a distribution against alternatives separated in sup norm. We study the twin settings of Poisson-generated count data with a large number of categories and highdimensional multinomials. In previous studies of different separation metrics, it has been found that the local minimax separation rate exhibits substantial heterogeneity and is a complicated function of the null distribution; the rate-optimal test requires careful tailoring to the null. In the setting of sup norm, this remains the case and we establish that the local minimax separation rate is determined by the finer decay behavior of the category rates. The upper bound is obtained by a test involving the sample maximum, and the lower bound argument involves reducing the original heteroskedastic null to an auxiliary homoskedastic null determined by the decay of the rates. Further, in a particular asymptotic setup, the sharp constants are identified. This is a joint work with Subhodh Kotekal and Chao Gao.

04. Dezember 2024    

N. N.

11. Dezember 2024  

Chloe Rouyer (Universität Potsdam)

Foundations of Online Learning for Easy and Worst-case Data

Abstract: Online learning is a well-studied framework used to represent learning problems where the learner only has access to one data-point at the time and has to learn sequentially. This problem is particularly challenging in the bandit framework, which is a repeated game between the learner and the environment. In this game, the learner is faced with a list of actions and the environment generates losses associated with these actions. Then, the learner repeadly needs to play an action within this list in order to minimize their cumulative loss, but they can only observe the loss associated with the action they played. This means that at each round, the learner has to balance exploration (gathering information on less studied actions) and exploitation (using the already gathered information to play an action with a supposed small loss). Developing learner strategies for this problem depends on the assumptions made on the environment.

18. Dezember 2024   

 N. N.

08. Januar 2025

Johannes Schmidt-Hieber (University of Twente)

Statistical Estimation using Zeroth-Order Optimization

Abstract: In this talk, we study statistical properties of zeroth-order optimization schemes, which do not have access to the gradient of the loss and rely solely on evaluating the loss function. Such methods are often considered to be suboptimal for high-dimensional problems, as their convergence rates to the minimizer of the objective function are typically slower than those of gradient-based methods. This performance gap becomes more pronounced as the number of parameters increases.

Considering the linear model, we show that reusing the same data point for multiple zeroth-order updates can overcome the gap in the estimation rates. Additionally, we demonstrate that zeroth-order optimization methods can achieve the optimal estimation rate when only queries from the linear regression model are available. Special attention will be given to the non-standard minimax lower bound in the query model.

This is joint work with Thijs Bos, Niklas Dexheimer and Wouter Koolen.

15. Januar 2025

Xiaorui Zuo (NUS Singapore)

Cryptos have Rough Volatility and Correlated Jumps

Abstract: Contrary to expectations some years ago, the crypto market has matured and gives the impression of an established financial eco system. Certainly, some deviations from robustness, typically reflected in event related volatility bursts and spikes, are observed, but a liquid derivatives market has been established, at least for the dominant digital assets. It is therefore not only necessary for pricing contingent claims to understand the stochastic dynamics via a solid data analysis but also to provide instruments identifying volatility patterns and their dynamic evolvement. Using the Bitcoin as a representative instrument, we ventured to model this particular crypto coin dynamics via a combination of roughness in volatility and jumps in the underlying crypto currency. Findings on the roughness, e.g. the size of the Hurst exponent for the volatility dynamics, revealed remarkable differences when compared to corresponding estimates for equities and fixed income funds. Through a parametric bootstrap we give evidence that both roughness and jumps are crucial for predicting the range of next-day returns in terms of a simulated confidence interval. By scaling up the jump sizes we obtained a nicely working combination of volatility roughness and jumps (of the underlying) resulting in precise coverage levels. All calculations may be redone on quantlet.com and courselets are in quantinar.com

22. Januar 2025

Vincent Rivoirard (Université Dauphine, Paris)

PCA for point processes

Abstract: We introduce a novel statistical framework for the analysis of replicated point processes that allows for the study of point pattern variability at a population level. By treating point process realizations as random measures, we adopt a functional analysis perspective and propose a form of functional Principal Component Analysis (fPCA) for point processes. The originality of our method is to base our analysis on the cumulative mass functions of the random measures which gives us a direct and interpretable analysis. Key theoretical contributions include establishing a Karhunen-Loève expansion for the random measures and a Mercer Theorem for covariance measures. We establish convergence in a strong sense, and introduce the concept of principal measures, which can be seen as latent processes governing the dynamics of the observed point patterns. We propose an easy-to-implement estimation strategy of eigenelements for which parametric rates are achieved. We fully characterize the solutions of our  approach to Poisson and Hawkes processes  and validate our methodology via simulations and diverse applications in seismology, single-cell biology and neurosiences, demonstrating its versatility and effectiveness.

Joint work with Victor Panaretos (EPFL), Franck Picard (ENS de Lyon) and Angelina Roche (Université Paris Cité).

 

29. Januar 2025

Davy Paindaveine (Université libre de Bruxelles)

 

Rank tests for PCA under weak identifiability

 

Abstract: In a triangular array framework where n observations are randomly sampled from a p-dimensional elliptical distribution with shape matrix V_n, we consider the problem of testing the null hypothesis H_0: theta=theta_0, where theta is the (fixed) leading unit eigenvector of V_n and theta_0 is a given unit p-vector. The dependence of the shape matrix on the sample size allows us to consider challenging asymptotic scenarios in which the parameter of interest theta is unidentified in the limit, because the ratio between both leading eigenvalues of V_n converges to one. We study the corresponding limiting experiments under such weak identifiability, and we show that these may be LAN or non-LAN. While earlier work in the framework was strictly limited to Gaussian distributions, where the study of local log-likelihood ratios could simply rely on explicit expressions, our asymptotic investigation allows for essentially arbitrary elliptical distributions. Even in non-LAN experiments, our results enable us to investigate the asymptotic null and non-null properties of multivariate rank tests. These nonparametric tests are shown to exhibit an excellent behavior under weak identifiability: not only do they maintain the target nominal size irrespective of the amount of weak identifiability, but they also keep their uniform efficiency properties under such non-standard scenarios.

 

05. Februar 2025

Sophie Langer (University of Twente)

 

Deep learning theory – what’s next?

 

Abstract: 

In this talk, we delve into key theoretical breakthroughs, with a particular focus on statistical results. We critically question the prevailing frameworks and introduce a novel statistical approach to image analysis. Rather than treating images as high-dimensional data entities, our framework reconceptualized them as structured objects shaped by geometric deformations like shifts, scales, and orientations. The goal of the classification rule is then to learn the uninformative deformations, resulting in convergence rates with more favorable trade-offs between input dimension and sample size. This fresh perspective not only provides new guarantees for approximation and convergence in deep learning-based image classification but also redefines how we approach image analysis with the potential of broader applications to other learning tasks. We conclude by discussing emerging research directions and reflecting on the role of theory in the field.

This talk is based on joint work with Johannes Schmidt-Hieber and Juntong Chen.

 

12. Februar 2025

Judith Rousseau (University of Oxford / Paris Dauphine- PSL University)

 

Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions

 

Abstract: Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio and video generation as well as many more applications in science and beyond. The \textit{manifold hypothesis} states that high-dimensional data often lie on lower-dimensional manifolds within an ambient space of large  dimension D , and is widely believed to hold in provided examples. While recent results have provided invaluable insight into how diffusion models adapt to the manifold hypothesis, they do not capture the great empirical success of these models. 

In this work, we study DDPMs under the manifold hypothesis and prove that they achieve rates independent of the ambient dimension in terms of learning the score. In terms of sampling, we obtain rates independent of the ambient dimension w.r.t.\ the Kullback-Leibler divergence, and $O(\sqrt{D})$ w.r.t.\ the Wasserstein distance. We do this by developing a new framework connecting diffusion models to the well-studied theory of extrema of Gaussian Processes.

This is a joint work with I. Azangulov and  G. Deligliannidis (University of Oxford)

 Interessenten sind herzlich eingeladen.

Für Rückfragen wenden Sie sich bitte an:

Frau Marina Filatova

Mail: marina.filatova@hu-berlin.de
Telefon: +49-30-2093-45460
Humboldt-Universität zu Berlin
Institut für Mathematik
Unter den Linden 6
10099 Berlin, Germany