Fluctuations of generalized Fermat distances

• Grant number: 24/06341-9
• Support Opportunities: Scholarships abroad - Research
• Start date: October 22, 2024
• End date: July 26, 2025
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Probability
• Principal Investigator: Alexsandro Giacomo Grimbert Gallo
• Grantee: Alexsandro Giacomo Grimbert Gallo
• Host Investigator: Matthieu Jonckheere
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Institution abroad: Laboratoire d'Analyse et d'Architecture des Systèmes (LAAS), France
• Associated research grant: 23/13453-5 - Stochastic systems modeling, AP.TEM

Abstract

Main part. The main part, which gives the project its name, is the start of a new line of research for the proposer. It is a research project in probability theory (percolation, point processes, concentration inequalities, random graphs...) with motivations reaching theoretical machine learning problems (dimensionality reduction, clustering, topological data analysis...). Consider a set of n points generated by a not necessarily homogeneous point process on a subvariety of R^d. In a recent paper, Matthieu Jonckheere and collaborators proved, using first-passage percolation arguments, that the microscopic (sample) Fermat distance based on these n points converges almost surely to the macroscopic version when n diverges. The name Fermat distance comes from its relation to Fermat's optical principle. The relationship with theoretical machine learning is that inferring distances from data is an important step in certain tasks with high-dimensional data, such as clustering and dimensionality reduction.We have two goals in this part: (1) to obtain information about fluctuations in this convergence, and (2), to consider variations of this distance involving the ``distance to a measure''. Both problems will be developed with Matthieu Jonckheere, host, and Frédéric Chazal (INRIA-Saclay). Part attached. In parallel to the main project, we will continue ongoing projects on stochastic processes in discrete structures (Z^d, random trees and graphs) under their various approaches: recurrence theorems, statistical properties (concentration inequalities, limit theorems), percolation and inference.