Universität Wien

390026 SE PhD-AW: Seminar for Doctoral Candidates (2022W)

3.00 ECTS (2.00 SWS), SPL 39 - Doktoratsstudium Wirtschaftswissenschaften
Continuous assessment of course work
ON-SITE
Moodle

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Note: The time of your registration within the registration period has no effect on the allocation of places (no first come, first served).

Details

max. 10 participants
Language: English

Lecturers

Classes (iCal) - next class is marked with N

  • Monday 03.10. 15:00 - 16:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 04.10. 11:30 - 13:00 Seminarraum 15 Oskar-Morgenstern-Platz 1 3.Stock
  • Wednesday 05.10. 13:15 - 14:45 Hörsaal 8 Oskar-Morgenstern-Platz 1 1.Stock
  • Thursday 06.10. 11:30 - 13:00 Seminarraum 5 Oskar-Morgenstern-Platz 1 1.Stock
  • Friday 07.10. 15:00 - 16:30 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 07.10. 16:45 - 18:15 Digital
  • Thursday 02.02. 11:30 - 16:30 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 03.02. 09:45 - 13:00 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

Statistics beyond linear data October 3-7, 2022

Instructor: Victor-Emmanuel Brunel

This course aims at an introduction to probability and statistics on non-linear metric spaces. One of the main objectives of the course is to describe the interactions between the geometry of the metric space (e.g., its curvature properties) and data randomly generated in that space. The main object of study will be barycenters, a.k.a. Fréchet means, which extend the notion of arithmetic mean to non-linear metric spaces. We will introduce the main tools from metric and differential geometry that will allow to prove laws of large numbers and central limit theorems for barycenters.
After describing several examples of statistical problems where data lie in non-linear spaces and giving a general introduction on metric geometry and curvature, we will study barycenters: existence, uniqueness and statistical properties. We will then focus on the particular case of Riemannian manifolds, which provide an appropriate framework in order to prove central limit theorems.

Outline of the course:
Introduction
- Examples of data in non-linear spaces
- Different metric structures: from length spaces to Riemannian manifolds
- Short review of statistical theory for non-linear data

Geodesic spaces and curvature
- Length spaces, geodesic spaces
- Alexandrov's curvature: CAT spaces

Barycenters in CAT spaces
- Definition, existence, uniqueness and anomalies
- Consistency
- The case of non-positively curved spaces

A crash course on Riemannian geometry
- Smooth manifolds
- Riemmanian metrics, connections
- Exponential map and geodesics
- Parallel transport
- Sectional and Ricci curvatures

Statistics on Riemannian manifolds
- M-estimators and central limit theorems
- The case of barycenters and geometric medians
- Smeariness of empirical barycenters

Assessment and permitted materials

Minimum requirements and assessment criteria

Examination topics

Reading list

References:
Here is a non-exhaustive list of references that have inspired the contents of this course.

Metric geometry
1. Bridson, M. R. and Haefliger, A. (2013), Metric spaces of non-positive curvature,
Springer Science and Business Media.
2. Burago, D., Burago, Y. and Ivanov, S. (2022), A course in metric geometry, American Mathematical Society.
3. Ohta, S. I. (2007), Convexities of metric spaces, Geometriae Dedicata, 125(1), pp.
225-250.
4. Udriste, C. (2013), Convex functions and optimization methods on Riemannian
manifolds, Springer Science and Business Media.

Riemannian geometry
1. Lee, J. M. (2018), Introduction to Riemannian manifolds, Springer International
Publishing.
2. Do Carmo, M. P. (1992), Riemannian geometry, Birkhäuser Boston.

Barycenters in metric spaces
1. Sturm, K. T. (2003), Probability measures on spaces with nonpositive curvature.
Contemporary Mathematics Vol. 338, pp. 357-424.
2. Bhattacharya, R. and Patrangenaru, V. (2003), Large sample theory of intrinsic and
extrinsic sample means on manifolds - I, The Annals of Statistics, 31(1), pp. 1-29.
3. Bhattacharya, R. and Patrangenaru, V. (2005). Large sample theory of intrinsic
and extrinsic sample means on manifolds - I, The Annals of Statistics, 33(3), pp.
1225-1259.
4. Bhattacharya, R. and Lin, L. (2017), Omnibus CLTs for Fréchet means and nonparametric inference on non-Euclidean spaces. Proceedings of the American Mathematical Society, 145(1), pp. 413-428.
5. Karcher, H. (1977), Riemannian center of mass and mollifier smoothing, Communications on pure and applied mathematics, 30(5), pp. 509-541.
6. Ohta, S. I. (2012). Barycenters in Alexandrov spaces of curvature bounded below,
Advances in geometry, 12(4), 571-587.
7. Ohta, S. I. and Pálfia, M. (2015), Discrete-time gradient flows and law of large numbers in Alexandrov spaces, Calculus of Variations and Partial Differential Equations, 54(2), pp. 1591-1610.
8. Ahidar-Coutrix, A., Le Gouic, T. and Paris, Q. (2020), Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics, Probability theory and related fields, 177(1), pp. 323-368.
9. Le Gouic, T., Paris, Q., Rigollet, P. and Stromme, A. J. (2022), Fast convergence
of empirical barycenters in Alexandrov spaces and the Wasserstein space, Journal
of the European Mathematical Society.

Association in the course directory

Last modified: Th 11.05.2023 11:28