Deep learning architectures are becoming ubiquitous for inverse problem resolution. We start by reviewing one of the earliest examples. It consists in using recurrent neural networks to solve the Lasso problem: one of the most popular algorithm to solve the Lasso, the Iterative Shrinkage-Thresholding Algorithm (ISTA), iterates matrix multiplications and non-linearities until convergence. Therefore, T iterations of ISTA are equivalent to a T layers neural network. The weights of the neural network can then be learned to accelerate resolution of the Lasso. 
In a second step, we study the weights that are learned by such networks. In particular, we show that the last layers of such deep networks can only learn a better step-size for the ISTA iterations.

Optimal transport (OT) suffers from the curse of dimensionality. In this talk, I will present a new regularization of OT leveraging regularity of the Brenier map. Instead of considering regularity as a a property that can be proved under suitable assumptions, we consider regularity as a condition that must be enforced when estimating OT. From an algorithmic point of view, this leads to an infinite-dimensional optimization problem, which, when dealing with discrete measures, can be rewritten as a finite-dimensional separately convex problem. From a statistical point of view, this defines new estimators of the OT map and 2-Wasserstein distance between arbitrary measures, for which I will show numerical evidence of their performance.

(Based on a joint work with Alexandre d'Aspremont and Marco Cuturi)

Le séminaire aura lieu le 5 novembre 2019 de 16h00 à 17h30 au centre Inria-Saclay - bâtiment Alan Turing - Amphithéâtre Sophie Germain.

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