Optimization in inverse problems via inertial iterative regularization
Guillaume Garrigos - Université de Paris
n the context of linear inverse problems, we propose and study a general iterative regularization method allowing to consider large classes of regularizers and data-fit terms. We are particularly motivated by dealing with non-smooth data-fit terms, such like a Kullback-Liebler divergence, or an L1 distance. We treat these problems by studying both a continuous (ODE) and discrete (algorithm) dynamics, based on a primal-dual diagonal inertial method, designed to solve efficiently hierarchical optimization problems. The key point of our approach is that, in presence of noise, the number of iterations of our algorithm acts as a regularization parameter. In practice this means that the algorithm must be stopped after a certain number of iterations. This is what is called regularization by early stopping, an approach which gained in popularity in statistical learning. Our main results establishes convergence and optimal stability of our algorithm, in the sense that for additive data-fit terms we achieve the same rates than the Tikhonov regularisation method for linear problems.
Guillaume Garrigos has studied Applied Mathematics in Montpellier (France). He has obtained in 2015 a Franco-Chilean Ph.D. in Applied Mathematics from both Universite de Montpellier and Universidad Tecnica Federico Santa Maria, under the direction of Hedy Attouch and Juan Peypouquet. He then did a postdoc within the Laboratory for Computational and Statistical Learning, a joint lab between the IIT and the MIT, working in collaboration with Lorenzo Rosasco and Silvia Villa. After that, he joined Gabriel Peyre's team in the Ecole Normale Superieure de Paris, for a second postdoc. Since 2018, he is Maitre de Conferences (Associate Professor) at the Universite de Paris (formerly Paris-Diderot). His current research interests focus on the interplay between optimization and the regularization for inverse problems, arising in machine learning or in signal and image processing.
2019-05-22 at 02:30 pm