VAN DANTZIG SEMINAR

On Friday October 26, the Van Dantzig Seminar on statistics will host lectures by Gérard Kerkyacharian (Paris, Université Pierre et Marie Curie) and Stefan Sommer (University of Copenhagen)

The program is (titles and abstracts below):

14.30 – 15.30  Gérard Kerkyacharian
15.30 – 15.50. 10 break
15.50 – 16.50  Stefan Sommer
17.00 – drinks

Location: Delft University of Technology, EWI-building (highest building on the campus), lecture hall D@ta (Mekelweg 4).

Free attendance.

The Van Dantzig seminar is a nationwide series of lectures in statistics, that features renowned international and local speakers from the full breadth of the statistical sciences. The name honours David van Dantzig (1900-1959), who was the first modern statistician in the Netherlands, and professor in the “Theory of Collective Phenomena” (i.e. statistics) in Amsterdam. The seminar will convene 4 to 6 times a year at varying locations, and is financially supported by, among others, the STAR cluster and the Section Mathematical Statistics of the VVS-OR.

Everybody is cordially invited to attend.

TITLES AND ABSTRACTS OF PRESENTATIONS
Gerard Kerkyacharian

Regularity spaces, and wavelet in a geometric framework. Application to Gaussian processes and statistical estimation

We will show how, on a suitable Dirichlet space, one can define regularity spaces, and a wavelet system. This construction works for Riemannian manifolds, under some curvature condition, and relatively compact convex open domain of such Riemannian manifold. Then we revisit some classical results on the behavior of Gaussian processes, and non parametric adaptive statistical estimation.

1. T. Coulhon, G. Kerkyacharian, and P. Petrushev, Heat Kernel Generated Frames in the Setting of Dirichlet Spaces, J. Fourier Anal. Appl. 18 (2012), no. 5, 995-1066.

2. G. Kerkyacharian, P. Petrushev, Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces, Trans. Amer. Math. Soc. 367 (2015), no. 1, 121-189.

3. G. Kerkyacharian, S. Ogawa, P. Petrushev, D. Picard, Regularity of Gaussian processes on Dirichlet spaces, Constr. Approx. 47 (2018), no. 2, 277-320.

4. G. Cleanthous, A. Georgiadis, G. Kerkyacharian, P. Petrushev, D. Picard, Kernel and wavelet density estimators on manifolds and more general metric spaces. arXiv:1805.04682

5. G. Kerkyacharian, P.Petrushev, Yuan Xu, Gaussian bounds for the weighted heat kernels on the interval, ball and simplex. arXiv:1801.07325

Stefan Sommer

Probabilistic Approaches to Geometric Statistics

Geometric statistics, the statistical analysis of manifold and Lie group valued data, traditionally start with the Fréchet mean defined as a minimizer of the expected squared distance. Similar least-squares criterions are used to generalize regression and principal component analysis beyond the Euclidean situation. An alternative to least-squares is to optimize the data likelihood under families of parametric probability distributions on the nonlinear space. This probabilistic approach emphasizes maximum likelihood means over the Fréchet mean, and it allows generalization of Euclidean statistical procedures defined via the data likelihood to the manifold situation. While parametric families of probability distributions are generally hard to construct in nonlinear spaces, transition densities of stochastic processes provide a geometrically natural way of defining likelihoods. In the talk, I will discuss common least-squares constructions in geometric statistics and their probabilistic counterparts, construction of geometrically natural probability distributions, and how curvature couples with the probabilistic models in distinctly non-Euclidean ways. Simulation of manifold and Lie group valued diffusion bridges here play an integral role in evaluation of data likelihoods.

1. Stefan Sommer, An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data, Sankhya A, 2018, arXiv:1801.10341.
2. Alexis Arnaudon, Darryl D. Holm, and Stefan Sommer, A Geometric Framework for Stochastic Shape Analysis, Foundations of Computational Mathematics, 2018, arXiv:1703.09971.
3. Stefan Sommer, Alexis Arnaudon, Line Kuhnel, and Sarang Joshi, Bridge Simulation and Metric Estimation on Landmark Manifolds, Graphs in Biomedical Image Analysis, Computational Anatomy and Imaging Genetics, Lecture Notes in Computer Science, Springer, September 2017, pp. 79–91 (en).
4. Stefan Sommer and Anne Marie Svane, Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry, Journal of Geometric Mechanics 9 (2017), no. 3, 391–410 (en).