# Mathematical Statistics Seminar

## Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

Speaker(s):
Jakob Söhl (University of Cambridge)
Date:
Wednesday, June 22, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We consider nonparametric Bayesian inference in a reflected diffusion model $dX_t = b (X_t)dt + \sigma(X_t) dW_t,$ with discretely sampled observations $X_0, X_\Delta, \dots, X_{n\Delta}$. We analyse the nonlinear inverse problem corresponding to the low frequency sampling' regime where $\Delta>0$ is fixed and $n \to \infty$. A general theorem is proved that gives conditions for prior distributions $\Pi$ on the diffusion coefficient $\sigma$ and the drift function $b$ that ensure minimax optimal contraction rates of the posterior distribution over H\"older-Sobolev smoothness classes.

## Slope meets Lasso: improved oracle bounds and optimality

Speaker(s):
Pierre Bellec (CREST)
Date:
Wednesday, June 8, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We show that two polynomial time methods, a Lasso estimator with adaptively chosen tuning parameter and a Slope estimator, adaptively achieve the exact minimax prediction and \ell_2 estimation rate (s/n) log(p/s) in high-dimensional linear regression on the class of s-sparse target vectors in \mathbb{R}^p. This is done under the Restricted Eigenvalue (RE) condition for the Lasso and under a slightly more constraining assumption on the design for the Slope. The main results have the form of sharp oracle inequalities accounting for the model misspecification error.

## Nonparametric minimax tests for large covariance matrices - CANCELLED!

Speaker(s):
Cristina Butucea (Marne-la-Vallée)
Date:
Wednesday, June 1, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We observe n independent p-dimensional Gaussian vectors with missing values, that is each coordinate (which is assumed standardized) is observed with probability a>0. Asymptotically, n and p tend to infinity, a tends to 0. We investigate the test problem of a simple null hypothesis that the high-dimensional covariance matrix of the underlying random vector is the identity matrix (lack of correlations).

## Structured Semi-Definite Programming with Applications to Non-Gaussian Component Analysis

Speaker(s):
Yury Maximov (IITP RAS, Moscow)
Date:
Wednesday, May 25, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Semi-definite programming (SDP) is a popular tool for approximation of non-convex quadratic problems arises in multiple statistical and computer science problems. Known to be worst-case optimal SDP is often dominated on well-structured (practical) problems by domain specific methods and heuristics. Yet another problem of SDP is a slow computational time makes it hardly applicable for huge-scale problems. In this talk we try to incorporate problem structure in the semi-definite dual to contribute both decrease computational time and improve approximation guarantees.

## Dictionary learning: principles, algorithms, guarantee

Speaker(s):
Rémi Gribonval (INRIA Rennes)
Date:
Wednesday, May 18, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Sparse modeling has become highly popular in signal processing and machine learning, where many tasks can be expressed as under-determined linear inverse problems. Together with a growing family of low-dimensional signal models, sparse models expressed with signal dictionaries have given rise to a rich set of algorithmic principles combining provably good performance with bounded complexity. In practice, from denoising to inpainting and super-resolution, applications require choosing a “good” dictionary.

## Oracle Estimation of a Change Point in High Dimensional Quantile Regression

Speaker(s):
Simon Lee (Seoul National University)
Date:
Wednesday, May 11, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

In this paper, we consider a high dimensional quantile regression model where the sparsity structure may differ between the two sub-populations.We develop 1-penalized estimators of both regression coefficients and the threshold parameter. Our penalized estimators not only select covariates but also discriminate between a model with homogeneous sparsity and a model with a change point. As a result, it is not necessary to know or pretest whether the change point is present, or where it occurs.

## Bootstrap tuned model selection

Speaker(s):
Date:
Wednesday, May 4, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

In the problem of model selection for a given family of linear estimators \tilde{\theta}_m, m \in, ordered by their variance, we offer a new "smallest accepted" approach motivated by Lepski's device and the multiple testing idea. The procedure selects the smallest model which satisfies the acceptance rule based on comparison with all larger models.

## Is adaptive early stopping possible in statistical inverse problems?

Speaker(s):
Gilles Blanchard (Universität Potsdam)
Markus Reiß (Humboldt-Universität zu Berlin)
Date:
Wednesday, April 27, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We consider a standard setting of statistical inverse problem, taking the form of the Gaussian sequence model with D observed noisy coefficients. Consider the simple family of keep or kill estimators depending on a cutoff index k_0.

## Monge-Kantorovich Ranks and Signs

Speaker(s):
Marc Hallin (de L'Université libre de Bruxelles)
Date:
Wednesday, April 20, 2016 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Unlike the real line, the real space RK, K≥ 2 is not `naturally" ordered. As a consequence, such fundamentals univariate concepts as quantile and distribution functions, ranks, signs, all order-related, do not straightforwardly extend to the multivariate context. Since no universal pre-existing order exists, each distribution, each data set, has to generate its own---the rankings behind sensible concepts of multivariate quantile, ranks, or signs, inherently will be distribution-specific and, in empirical situations, data-driven.

Speaker(s):