Vladimir Koltchinskii
Vladimir Koltchinskii is a professor in Mathematics at Georgia Tech. His current research is primarily in high-dimensional statistics and probability.
Title
Estimation of Functionals of High-Dimensional and Infinite-Dimensional Parameters of Statistical Models: Bias Reduction and Concentration
Abstract
This mini-course deals with a circle of problems related to estimation of real valued functionals of high-dimensional and infinite-dimensional parameters of statistical models. In such problems, it is of interest to estimate one-dimensional features of a high-dimensional parameter represented by nonlinear functionals of certain degree of smoothness defined on the parameter space. The functionals of interest could be often estimated with faster convergence rates than the whole parameter (sometimes, even with parametric rates).
We will discuss some mathematical methods providing a way to develop estimators of functionals of high-dimensional parameters with optimal error rates in classes of functionals of some H\"older smoothness and even to provide their efficient estimation with parametric rates when the smoothness is sufficiently large. The main focus will be on functionals of unknown covariance operators in high-dimensional and infinite-dimensional Gaussian models, where the functionals of interest often capture their spectral properties. In particular, we will discuss the role of higher order bias reduction methods and concentration inequalities in these problems.
The following topics will be discussed:
- Non-asymptotic bounds and concentration inequalities for sample covariance in high-dimensional
and dimension-free frameworks; - Some approaches to concentration inequalities for smooth functionals of statistical estimators;
- Higher order bias reduction methods in functional estimation (iterative bias reduction based on bootstrap chains; linear aggregation of plug-in estimators and jackknife estimators; methods based on Taylor expansion and estimation of polynomials with reduced bias);
- minimax lower bounds in functional estimation.
Bruno Loureiro
Bruno is a CNRS researcher based at the Centre for Data Science at the École Normale Supérieure in Paris working on the crossroads between machine learning and statistical mechanics. He also holds an Adjunct Professor (“Professeur Attaché”) position at the Université Paris Sciences et Lettres (PSL) where he teaches at the undergraduate and graduate programs of the affiliated universities.
Before moving to CNRS & ENS, he was a postdoc at the Institut de Physique Théorique (IPhT) in Paris and at the École Polytechnique Fédérale de Lausanne (EPFL) in Lausanne, where he worked with Lenka Zdeborová and Florent Krzakala.
Before his postdoc, he read the part III of the Mathematics Tripos at the University of Cambridge, and continued into a PhD at the TCM group in the same university. During his PhD he worked on the Holographic Principle, a duality stemming from string theory that relates quantum field theories to classical theories of gravity. His thesis was centered on applications of this duality to strongly coupled condensed matter systems, with a particular focus on disordered systems.
Although all of that sounds very different from his current research, it is funny to note that many of the methods he employed at the time are the same as in his current topic. In the end, Physics is all about Gaussian integrals, isn’t it?
To find out more about Bruno's research and activities, you may visit his website here.
Potential topic
Statistical physics tools for high-dimensional learning problems.
Cynthia Rush
Cynthia Rush is an Associate Professor of Statistics in the Department of Statistics at Columbia University. She received her Ph.D. in Statistics from Yale University in 2016, under the supervision of Andrew Barron. She obtained her B.S. in Mathematics at the University of North Carolina at Chapel Hill.
Her research uses tools and ideas from information theory, statistical physics, and applied probability as a framework for understanding modern, high-dimensional inference and estimation problems and complex machine learning tasks that are core challenges in statistics and data science.
To find out more about Cynthia's research and activities, you may visit her website here.
Potential topic
High dimensional statistics and approximate message passing.
Matus Telgarsky
Matus Telgarsky is an Assistant Professor at the Courant Institute of Mathematical Sciences at New York University, specializing in deep learning theory. He was fortunate to receive a PhD at UCSD under Sanjoy Dasgupta. Other highlights include: co-founding, in 2017, the Midwest ML Symposium (MMLS) with Po-Ling Loh (while on faculty at the University of Illinois, Urbana-Champaign); receiving a 2018 NSF CAREER award; and organizing two Simons Institute programs, one on deep learning theory (summer 2019), and one on generalization (fall 2024).
To find out more about Matus' research and activities, you may visit his website here.
Potential topic
Neural networks theory.