Topological Data Analysis: Statistical Methods and Inference – TopData

TopData stands for Topological Data Analysis: Statistical Methods and Inference. TopData aims at designing new mathematical frameworks, models and algorithmic tools to infer and analyze the topological and geometric structure of data in different statistical settings. Its goal is to set up the mathematical and algorithmic foundations of Statistical Topological and Geometric Data Analysis and to provide robust and efficient tools to explore, infer and exploit the underlying geometric structure of various data.

Our conviction, at the root of this project, is that there is a real need to combine statistical and topological/geometric approaches in a common framework, in order to face the challenges raised by the inference and the study of topological and geometric properties of the wide variety of larger and larger available data.
We are also convinced that these challenges need to be addressed both from the mathematical side and the algorithmic and application sides.
Our project brings together in a unique way experts in Statistics, Geometric Inference and Computational Topology and Geometry. Our common objective is to design new theoretical frameworks and algorithmic tools and thus to contribute to the emergence of a new field at the crossroads of these domains. Beyond the purely scientific aspects we hope this project will help to give birth to an active interdisciplinary community. With these goals in mind we intend to promote, disseminate and make our tools available and useful for a broad audience, including people from other fields.

Our research project is divided into five tasks.
1. Geometric inference and persistent homology for metric spaces.
2. Inference of measures.
3. Statistical methods for geometric inference.
4.Data structures and algorithms for topological data analysis.
5. Software, benchmark datasets and dissemination.

The first three tasks address problems from the mathematical point of view and bring statistical, probabilistic, topological and geometric techniques together to derive new methods for the analysis of data.
Task 4 focuses on the algorithmic aspects of the project while Task 5 is dedicated to the technological outcomes of our project and their disseminations.

Overall 14 permanent researchers, 4 PhD students (not funded by ANR) and 1 engineer (not funded by ANR) are participating to this proposal. Each partner will hire a one year post-doctoral researcher (funded by the ANR) to work on the project. We propose to get organized around four main partners:
- INRIA Saclay, which gathers statisticians from the Mathematics Lab at Université Paris-Sud Orsay and applied topologists and geometers from INRIA.
- LSTA at Université Paris 6, which gathers statisticians from various universities in the center of Paris.
- INRIA Sophia-Antipolis, which brings in experts in Computational Geometry.
- GIPSA-LAB in Grenoble, which brings in experts in Computational Topology.