2022 - T3 - WS2 - Geometry, Topology and Statistics in Data Sciences

Collection 2022 - T3 - WS2 - Geometry, Topology and Statistics in Data Sciences

Organizer(s) Aamari, Eddie ; Aaron, Catherine ; Chazal, Frédéric ; Fischer, Aurélie ; Hoffmann, Marc ; Le Brigant, Alice ; Levrard, Clément ; Michel, Bertrand

Date(s) 10/10/2022 - 14/10/2022

linked URL https://indico.math.cnrs.fr/event/7546/

4 15

Reconstructing manifolds by weighted L1-norm minimization

By Dominique Attali

In many practical situations, the shape of interest is only known through a finite set of data points. Given as input those data points, it is then natural to try to construct a triangulation of the shape, that is, a set of simplices whose union is homeomorphic to the shape. This problem has given rise to many research works in the computational geometry community, motivated by applications to 3D model reconstruction and manifold learning.

In this talk, we focus on one particular instance of the shape reconstruction problem, in which the shape we wish to reconstruct is an orientable smooth d-dimensional submanifold of the $N$-dimensional Euclidean space. We reformulate the problem of searching for a triangulation as a convex minimization problem, whose objective function is a weighted L1-norm. I will then present a result which says that, under appropriate conditions, the solution of our minimization problem is indeed a triangulation of the manifold and that this triangulation coincides with a variant of the tangential Delaunay complex.

This is a joint work with André Lieutier.

Information about the video

Date of recording 13/10/2022
Date of publication 06/05/2024
Institution IHP
Language English
Audience Researchers, Graduate Students, General Public, Students
Director(s) Alexandre Duplessis, Marco Perez
Format MP4
Venue IHP - Hermite Amphitheater

Citation data

DOI 10.57987/IHP.2022.T3.WS2.004
Cite this video Attali, Dominique (13/10/2022). Reconstructing manifolds by weighted L1-norm minimization. IHP. Audiovisual resource. DOI: 10.57987/IHP.2022.T3.WS2.004
URL https://dx.doi.org/10.57987/IHP.2022.T3.WS2.004

Domain(s)

Computational Geometry

Bibliography

Dominique Attali, André Lieutier / Flat Delaunay complexes for homeomorphic manifold reconstruction. 2022 arXIv:2203.05943
Dominique Attali, André Lieutier / Delaunay-like Triangulation of Smooth Orientable Submanifolds by L1-Norm Minimization. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry (SoCG 2022), volume 224 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1– 8:16, Dagstuhl, Germany, 2022. Schloss Dagstuhl–Leibniz-Zentrum für Informatik arXIv:2203.06008

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