CE40 - Mathématiques

Geometry in Data: Statistical Inference & Clustering – GeoDSIC

Submission summary

The mathematics of data science concern several fields in mathematics, including statistics and geometry. This project is dedicated to geometric and topological statistical inference, which raises important mathematical challenges. Given an unknown shape, the goal is to learn some of its features based on observations. Often, the data consists of a point cloud sampled on the shape, possibly corrupted by noise. More generally, observing d variables on n individuals, one tries to grasp a nonlinear relation between variables, which turns out to lie in a shape with smaller dimension. We will study questions ranging from clustering and quantization to estimation of a shape. A key point will be the quest for robustness to noise. We will build adaptive methods and develop tractable procedures with theoretical guarantees.
Our project encompasses 5 topics.
The first one gathers several problems in manifold estimation. In particular, we focus on noisy data, and on estimating the reach, regularity parameter of a manifold. We also consider the time-dependent setting.
A second subject deals with constrained principal curves and surfaces, in estimation and statistical learning, including the algorithmic aspect.
Then, the quantization and clustering topic includes the study of these methods on quantum graphs and the exploration of an aggregation of partitions based on spectral clustering.
Another theme concerns questions involving Wasserstein metrics, in deconvolution, but also in the context of estimation using generative adversarial networks.
Finally, the last topic relates to topological data analysis, with the construction of a nonparametric statistical test, as well as software implementation perspectives.

Project coordination

Aurélie Fischer (Sorbonne Université)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

LMJL Ecole Centrale de Nantes
LPSM Sorbonne Université

Help of the ANR 316,607 euros
Beginning and duration of the scientific project: December 2022 - 60 Months

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