Bo\'az Klartag (The Weizmann Institute of Science)
Isoperimetric inequalities in high-dimensional convex sets
10:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Prof. Dr. Marco Mondelli (ISTA)
Precise Asymptotics for Spectral Estimators: A Story of Phase Transitions, Random Matrices and Approximate Message Passing abstract
Abstract:
Spectral methods are a simple yet effective approach to extract information from data which is high-dimensional, i.e., where sample size and signal dimension grow proportionally. As a prelude, we will consider the prototypical problem of inference from a generalized linear model with an i.i.d. Gaussian design. Here, the spectral estimator is the principal eigenvector of a data-dependent matrix. We will discuss the emergence of a (BBP-like) phase transition in the spectrum of this random matrix and how such phase transition is related to signal recovery. The core of the talk will then deal with two models that capture the heterogeneous and structured nature of practical data. First, we will consider a multi-index model where the output depends on the inner product between the feature vector and a fixed number $p$ of signals, and the focus is on recovering the subspace spanned by the signals via spectral estimators. By using tools from random matrix theory, we will locate the top-$p$ eigenvalues of the spectral matrix and establish the overlaps between the corresponding eigenvectors (which give the spectral estimators) and a basis of the signal subspace. Second, we will consider a generalized linear model with a correlated design matrix. Here, the analysis of the spectral estimator relies on tools based on approximate message passing, and we will present a methodology which is broadly applicable to the study of spiked matrices. In all these settings, the precise asymptotic characterization we put forward enables the optimization of the data preprocessing, thus allowing to identify the spectral estimator that requires the minimal sample size for signal recovery.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Sasha Merkurjev
Title T.B.A.
14:15 • EPF Lausanne
Léo Belzile
tba
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Susan Hermiller (University of Nebraska)
Unknotting number is not additive under connected sum
Fabian Ziltener (ETH Zürich)
Variationsrechnung, Symmetrien und Erhaltungssätze
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Prof. Dr. Jonathan Ziveyi (University of New South Wales)
Title T.B.A.
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Mani Esna Ashari
17:30 • Universität Basel, Departement Mathematik und Informatik, Spiegelgasse 5, Basel, Seminarraum 05.002