Geometry of sample spaces

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningfagfællebedømt

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of Mn modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov–Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fréchet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

TidsskriftDifferential Geometry and its Application
Antal sider29
StatusUdgivet - 2023

Bibliografisk note

Funding Information:
The authors would like to thank François-Xavier Vialard for helpful discussions. P. Harms was funded by the National Research Foundation Singapore under the award NRF-NRFF13-2021-0012 and by Nanyang Technological University Singapore under the award NAP-SUG . X. Pennec was funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement Nr. 786854 G-Statistics). He was also supported by the French government through the 3IA Côte d'Azur Investments ANR-19-P3IA-0002 managed by the French National Research Agency (ANR). S. Sommer is supported by the Villum Foundation Grants 40582 and the Novo Nordisk Foundation grant NNF18OC0052000 .

Publisher Copyright:
© 2023 Elsevier B.V.


ID: 358549691