# A05 – Combining non parametric statistical and probabilistic approaches for inference on cloud-of-points data

This project terminated in June 2021 and will not be continued in the second funding period.

## Objectives

This project combines non-parametric statistical and probabilistic approaches for inference on cloud-of-points data. Such data sets can be modeled as point processes, which can be embedded into Hilbert spaces using appropriate reproducing kernels. The project investigates convergence rates for the associated non-parametric estimators and extends available results to probabilistic point process models with interaction, such as Gibbs fields and point processes over path spaces, e.g. "clouds of trajectories".

The research programme within this project follows first two major directions:

1-Analyze the convergence rate of Reproducing Kernel Hilbert Space methods over a generic metric space and identify the role of key quantities (metric entropy, distribution characteristics of covariate X such as probabilities of small balls)

2- Link to point process theory: focus on point processes with generating distributions of Gibbsian type and construct kernels taking into account specific properties. Example: One considers an unknown but fixed Gibbs interaction potential with random activity (corresponding to the reference density of points in the cloud)

## Preprints

Houdebert, P., Zass, A. (2020),

*An explicit continuum Dobrushin uniqueness criterion for Gibbs point processes with non-negative pair potentials*. arxiv 2009.06352Houdebert, P. (2019).

*Phase transition of the non-symmetric Continuum Potts model*. arXiv: 1908.10066Gribonval, R., Blanchard, G., Keriven, N. and Traonmilin, Y. (2017).

*Compressive Statistical Learning with Random Feature Moments.*arXiv 1706.07180

## Publications

Blanchard, G., Deshmukh, A., Dogan, U., Lee, G. and Scott, C. (2021).

*Domain Generalization by Marginal Transfer Learning*. Journal of Machine Learning Research 22(2):1−55. Open AccessZass, A. (2020).

*A Gibbs point process of diffusions: existence and uniqueness.*Proceedings of the XI international conference stochastic and analytic methods in mathematical physics (Lectures in pure and applied mathematics 6), Universitätsverlag Potsdam, 13-22. Open AccessHoudebert, P. (2020).

*Numerical study for thephase transition of thearea-interaction model.*Proceedings of the XI international conference stochastic and analytic methods in mathematical physics (Lectures in pure and applied mathematics 6), Universitätsverlag Potsdam, 165–174. Open AccessRoelly, S. and Zass, A. (2020).

*Marked Gibbs Point Processes with Unbounded Interaction: An Existence Result.*Journal of Statistical Physics 179, 972–996 (2020). Open AccessKatz-Samuels, J., Blanchard, G. and Scott, C. (2019).

*Decontamination of Mutual Contamination Models.*Journal of Machine Learning Research (41):1−57, 2019 Open AccessBlanchard, G. and Mücke, N. (2018).

*Parallelizing Spectral Algorithms for Kernel Learning.*Journal of Machine Learning Research (30):1-29, 2018. Open Access