Collection 2022 - T3 - WS1 - Non-Linear and High Dimensional Inference

Organizer(s) Aamari, Eddie ; Aaron, Catherine ; Chazal, Frédéric ; Fischer, Aurélie ; Hoffmann, Marc ; Le Brigant, Alice ; Levrard, Clément ; Michel, Bertrand

Date(s) 03/10/2022 - 07/10/2022

linked URL https://indico.math.cnrs.fr/event/7545/

11 21

Stein effect for estimating many vector means: a "blessing of dimensionality" phenomenon

By Gilles Blanchard

Consider the problem of joint estimation of the means for a large number of distributions in $R^d$ using separate, independent data sets from each of them, sometimes also called "multi-task averaging" problem. We propose an improved estimator (compared to the naive empirical means of each data set) to exploit possible similarities between means, without any related information being known in advance. First, for each data set, similar or neighboring means are determined from the data by multiple testing. Then each naive estimator is shrunk towards the local average of its neighbors. We prove that this approach provides a reduction in mean squared error that can be significant when the (effective) dimensionality of the data is large, and when the unknown means exhibit structure such as clustering or concentration on a low-dimensional set. This is directly linked to the fact that the separation distance for testing is smaller than the estimation error in high dimension and generalizes the well-known James-Stein phenomenon. An application of this approach is the estimation of multiple kernel mean embeddings, which plays an important role in many modern applications.

(This is based on joined work with Hannah Marienwald and Jean-Baptiste Fermanian)

Information about the video

Date of recording 06/10/2022
Date of publication 16/04/2024
Institution IHP
Language English
Audience Researchers, Graduate Students
Director(s) Alexandre Duplessis, Marco Perez
Format MP4
Venue Amphitheater Hermite, IHP

Citation data

DOI 10.57987/IHP.2022.T3.WS1.011
Cite this video Blanchard, Gilles (06/10/2022). Stein effect for estimating many vector means: a "blessing of dimensionality" phenomenon. IHP. Audiovisual resource. DOI: 10.57987/IHP.2022.T3.WS1.011
URL https://dx.doi.org/10.57987/IHP.2022.T3.WS1.011

Domain(s)

Statistics Theory

Bibliography

H. Marienwald, J-B. Fermanian, G. Blanchard / High-Dimensional Multi-Task Averaging and Application to Kernel Mean Embedding. Artificial Intelligence and Statistics (AISTATS 2021)