Date : Dec. 20, 2023, 3 p.m. - Tien Tam TRAN - Amphi Bruno Garcia
Constrained and Low Rank Gaussian Process on Some Manifolds
Gaussian processes have been widely and successfully used in machine learning applications. In this thesis, we propose to solve some limitations due to complexity and efficiency in high dimensions. Hence, we first investigate low complexity Gaussian processes by exploiting the Karhunen-Loève expansions. The proposed models reduce the computational cost, capture complex patterns, and maintain numerical efficiency. Second, we introduce a novel approach that employs Gaussian processes with hard constraints to approximate any density function. We propose a new formulation to simplify the geometric structure of the underlying space. As a result, the posterior distribution is normal constrained to the unit sphere, and it is numerically approximated using Hamiltonian Monte Carlo. The effectiveness of the proposed framework is validated through a series of various and multiple experiments. Finally, we introduce transfer learning models in the space of finite probability measures. We provide theoretical guaranties as well as detailed algorithms for numerical implementations. Consequently, we were able to generalize the transfer of Principal Components for Analysis (PCA) and linear regression models. Again, we illustrate and evaluate the proposed methods through various experiments.
This PhD was funded by the Agence National de la Recherche (ANR).
The jury composed by:
Pr. Christophette BLANCHET-SCALLIET, l’École Centrale de Lyon, France , Reviewer
Pr. Hong-Van LE, Czech Academy of Sciences, Czech Republic, Reviewer
Pr. Shantanu H. JOSHI, University of California Los Angeles, USA, Member
Pr. Rodolphe LE RICHE, Director CNRS, France, Member
Pr. Mourad BAIOU, Director CNRS, France, Member
Pr. José BRAGA, University Paul Sabatier, France, Invited
Pr. Chafik SAMIR, University of Clermont Auvergne, France, Advisor.