Charles Arnal

I am a postdoctoral researcher at Inria and the Laboratoire de Mathématiques d'Orsay, Paris-Saclay.
Before that, I completed a PhD under the supervision of Ilia Itenberg, at the IMJ-PRG, Sorbonne Université.
My research interests include Real Algebraic Geometry, Machine Learning in general and Topological Data Analysis in particular, and any topic at the intersection of Geometry and Statistics.

Publications

Real algebraic geometry

Patchworking real algebraic hypersurfaces with asymptotically large Betti numbers, in the Journal of Topology, 2022.
Lefschetz section theorems for tropical hypersurfaces, with Arthur Renaudineau and Kristin Shaw, in the Annales Henri Lebesgue, 2021.
Complex conjugation and simplicial algebraic hypersurfaces, arXiv preprint arXiv:2105.12100, 2021.

Topological data analysis and computational geometry

Critical points of the distance function to a generic submanifold, with Vincent Divol and David Cohen-Steiner, arXiv preprint arXiv:2312.13147, 2024.
Wasserstein convergence of Čech persistence diagrams for samplings of submanifolds, with Vincent Divol and David Cohen-Steiner, Hal preprint hal-04617508, 2024.
The distance function to a finite set is a topological Morse function, arxiv preprint arxiv:2407.15578, 2024.

Statistics and machine learning

Prompt Selection Matters: Enhancing Text Annotations for Social Sciences with Large Language Models, with Louis Abraham and Antoine Marie, arXiv preprint arxiv:2407.10645, 2024.
Iteration Head: A Mechanistic Study of Chain-of-Thought, with Vivien Cabannes, Wassim Bouaziz, Alice Yang, Francois Charton and Julia Kempe, arXiv preprint arxiv:2406.02128, 2024.
Mode Estimation with Partial Feedback, with Vivien Cabannes and Vianney Perchet, in COLT, 2024.
Touring sampling with pushforward maps, with Vivien Cabannes, in ICASSP, 2024 (Best Industry Paper award).
MAGDiff: Covariate Data Set Shift Detection via Activation Graphs of Deep Neural Networks, with Felix Hensel, Mathieu Carrière, Théo Lacombe, Hiroaki Kurihara, Yuichi Ike and Frédéric Chazal, in the Transactions on Machine Learning Research, 2024.
Convolution of a symmetric log-concave distribution and a symmetric bimodal distribution can have any number of modes, in the Statistics & Probability Letters, 2021.

PhD thesis

Patchworking, tropical homology, and Betti numbers of real algebraic hypersurfaces