Sara van de Geer is Professor of Statistics in the Department of Mathematics, ETH Zürich. Her work focuses on mathematical statistics, for example, theory for M-estimators in high/infinite dimensions, adaptation to unknown sparsity, semi-parametric theory, confidence sets in high-dimensional models, and concentration of measure for high-dimensional and nonparametric problems. She has (co-)authored four monographs, most recently lecture notes for the Saint-Flour Probability Summer School. She was a council member of the Swiss National Science Foundation 2007–2015, and is President of the Bernoulli Society 2015-2017. She is a Knight in the Order of Orange-Nassau, a member of the German Academy of Sciences Leopoldina, and a correspondent of the Dutch Royal Academy of Sciences. Sara’s three Wald Lectures will be given at the World Congress in Toronto, on July 12, 14 and 15.

High-dimensional statistics: a triptych

High-dimensional statistics concerns the situation where the number of parameters $p$ is (much) larger than the number of observations $n$. This is quite common nowadays, and it has led to the development of new statistical methodology. These lectures present a selected overview of mathematical theory for sparsity inducing methods.

In the first lecture we will highlight the main ingredients for proving sharp oracle inequalities for regularized empirical risk minimizers. The regularization penalty will be taken to be a norm $\Omega$ on $p$-dimensional Euclidean space. Important is that the norm $\Omega$ is has a particular feature which we term the triangle property. We present as examples: the ${\ell }_1$-norm, norms generated from cones, the sorted ${\ell }_1$-norm, the nuclear norm for matrices and an extension to tensors. We then show sharp oracle inequalities for a broad class of loss functions.

The second lecture addresses the construction of asymptotic confidence intervals for parameters of interest. Here, we restrict ourselves to the linear and the graphical model. We prove asymptotic normality of de-biased estimators. We consider asymptotic lower bounds for the variance of an approximately unbiased estimator of a one-dimensional parameter as well as Le Cam-type lower bounds. We ascertain the approximate unbiasedness of the de-biased estimator under sparsity conditions and show that it reaches the lower bound.

In the third lecture, we examine the null space property for sparsity inducing norms. The null space property ensures exact recovery of certain sparsity patterns and is moreover a key ingredient for oracle results. We derive this property for the Gram matrix based on n copies of a $p$-dimensional random variable $X$, where we require moment conditions for finite dimensional projections of $X$ or the more general small ball property.

The lectures are based on joint work with Andreas Elsener, Jana Janková, Alan Muro and Benjamin Stucky.