# Mathematical Statistics Seminar

## Empirical Chaos Processes and their application in Blind Deconvolution

Speaker(s):
Felix Khramer (TU München)
Date:
Wednesday, January 27, 2016 - 10:00am
Location:
WIAS, Raum 4.13, Hausvogteiplatz 11a, 10117 Berlin

The motivation of this talk is the deconvolution of two unknown vectors $w$ and $x$, each of which is sparse with respect to a generic (but known) basis. That is, one seeks to recover $w$ and $x$ from their circular convolution $y = w {\ast} x$. In this talk, we prove a restricted isometry property for this problem, which then entails convergence guarantees for the non-convex sparse power factorization algorithm via recent work by Lee et al.

## Robust and nonparametric detection of shifts using two-sample U-statistics and U-quantiles

Speaker(s):
Roland Fried (TU Dortmund)
Date:
Wednesday, January 13, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Tests for detecting level shifts in near epoch dependent time series are studied. The popular CUSUM test is not robust to outliers and can be improved in case of non-normal data, particularly for heavy-tails. The CUSUM test can be modified using the Hodges-Lehmann 2-sample estimator, which is the median of all pairwise differences between the samples. It is highly robust and has a high efficiency under normality.

## Asymptotic distribution of some robust and non-parametric change-point tests for time series

Speaker(s):
Herold Dehling (Ruhr-Universität Bochum)
Date:
Wednesday, January 13, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

In my talk, I will present recent results on the asymptotic distribution of some robust and non-parametric test statistics for the detection of change-points in time series. This leads to the study of two-sample U-processes, empirical U-processes, and two-sample U-quantiles.We will present limit theorems for these processes, both in the case of short range as well as long range dependent data.
(Joint work with Roland Fried, Martin Wendler, Murad Taqqu, Aeneas Rooch, and Isabel Garcia)

## Distribution of Linear Statistics of Singular Values of the Product of Random Matrices

Speaker(s):
Alexey Naumov (Moscow State University)
Date:
Wednesday, January 6, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

In my talk I will consider the products of random matrices. This topic is one of the most active in Random matrix theory in the last five years. I will be mostly interested in the limiting behavior of eigenvalues and singular values of such matrices. In particular, I will prove the central limit theorem for linear statistics of singular values. Some applications to statistics and theory of telecommunications will be discussed as well. This talk is based on the joint results with F. Goetze and A. Tikhomirov.

## Concentration bounds and asympotic distribution for the empirical spectral projectors of sample covariance operators

Speaker(s):
Karim Lounici (Georgia Institute of Technology, Atlanta)
Date:
Wednesday, December 9, 2015 - 10:00am
Location:
Paul-Drude-Institut für Festkörperelektronik, Hausvogteiplatz 5-7, 10117 Berlin, EG Raum 007

Let $X,X_1,\dots, X_n$ be i.i.d.

## Non-asymptotic upper bounds for the reconstruction error of PCA

Speaker(s):
Martin Wahl (HU Berlin)
Date:
Wednesday, December 2, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Principal component analysis (PCA) is a standard tool for dimension reduction. Classical objects of interest are the principal subspaces and their empirical counterparts. In this talk, we focus on the reconstruction error of PCA, and prove a non-asymptotic upper bound for the corresponding excess risk. This bound unifies and improves several upper bounds which were previously obtained by empirical process theory. Moreover, the bound reveals that the excess risk differs considerably from the usual subspace distance based on canonical angles.

## On the Optimality of Averaging in Distributed Statistical Learning

Speaker(s):
Jonathan Rosenblatt (Ben-Gourion University Negev)
Date:
Wednesday, November 25, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

A common approach to statistical learning on big data is to randomly split it among m machines and calculate the parameter of interest by averaging their m individual estimates. Focusing on empirical risk minimization, or equivalently M-estimation, we study the statistical error incurred by this strategy. We consider two asymptotic settings: one where the number of samples per machine n \to inf but the number of parameters p is fixed, and a second high-dimensional regime where both p, n \to inf with p/n \to \kappa.

## Fast low-rank estimation by projected gradient descent: Statistical and algorithmic guarantees

Speaker(s):
Martin Wainwright (University Berkeley)
Date:
Wednesday, October 28, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the low-rank matrix, and to run projected gradient descent on the nonconvex problem in the lower dimensional factorized space. We provide a general set of conditions under which projected gradient descent, when given a suitable initialization, converges geometrically to a statistically useful solution.

## Optimal Sup-norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Regression

Speaker(s):
Xiaohong Chen (Yale University)
Date:
Wednesday, October 21, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

This talk makes several contributions to the literature on the important yet difficult problem of estimating functions nonparametrically using instrumental variables. First, we derive the minimax optimal sup-norm convergence rates for nonparametric instrumental variables (NPIV) estimation of the structural function h_0 and its derivatives. Second, we show that a computationally simple sieve NPIV estimator can attain the optimal sup-norm rates for h_0 and its derivatives when h_0 is approximated via a spline or wavelet sieve.

## Testing the Specification in Random Coefficient Models

Speaker(s):
Christoph Breunig (Humboldt-Unverisität zu Berlin)
Date:
Wednesday, July 8, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

In this talk, we suggest and analyze a new class of specification tests for random coefficient models. They allow to assess the validity of central structural features of the model, in particular linearity in coefficients, generalizations of this notion like a known nonlinear functional relationship, or degeneracy of the distribution of a random coefficient, i.e., whether a coefficient is fixed or random, including whether an associated variable can be omitted altogether. Our tests are nonparametric in nature, and use sieve estimators of the characteristic function.