Publications associées à Banach fixed-point theorem

Stable parameterization of continuous and piecewise-linear functions

Michaël Unser, Alexis Marie Frederic Goujon, Joaquim Gonçalves Garcia Barreto Campos

Rectified-linear-unit (ReLU) neural networks, which play a prominent role in deep learning, generate continuous and piecewise-linear (CPWL) functions. While they provide a powerful parametric representation, the mapping between the parameter and function s ...

2023

The exploration process of critical Boltzmann planar maps decorated by a triangular O(n) loop model

Aleksandra Korzhenkova

In this paper we investigate pointed (q, g, n)-Boltzmann loop-decorated maps with loops traversing only inner triangular faces. Using peeling exploration Budd (2018) modified to this setting we show that its law in the non-generic critical phase can be cod ...

IMPA2022

Controlling the Complexity and Lipschitz Constant improves Polynomial Nets

Volkan Cevher, Grigorios Chrysos, Zhenyu Zhu, Fabian Ricardo Latorre Gomez

While the class of Polynomial Nets demonstrates comparable performance to neural networks (NN), it currently has neither theoretical generalization characterization nor robustness guarantees. To this end, we derive new complexity bounds for the set of Coup ...

2022

Predicting in Uncertain Environments: Methods for Robust Machine Learning

Paul Thierry Yves Rolland

One of the main goal of Artificial Intelligence is to develop models capable of providing valuable predictions in real-world environments. In particular, Machine Learning (ML) seeks to design such models by learning from examples coming from this same envi ...

EPFL2022

Learning Lipschitz-Controlled Activation Functions in Neural Networks for Plug-and-Play Image Reconstruction Methods

Michaël Unser, Dimitris Perdios, Pakshal Narendra Bohra, Alexis Marie Frederic Goujon, Sébastien Alexandre Emery

Ill-posed linear inverse problems are frequently encountered in image reconstruction tasks. Image reconstruction methods that combine the Plug-and-Play (PnP) priors framework with convolutional neural network (CNN) based denoisers have shown impressive per ...

2021

Lipschitz constant estimation for Neural Networks via sparse polynomial optimization

Volkan Cevher, Paul Thierry Yves Rolland, Fabian Ricardo Latorre Gomez

We introduce LiPopt, a polynomial optimization framework for computing increasingly tighter upper bounds on the Lipschitz constant of neural networks. The underlying optimization problems boil down to either linear (LP) or semidefinite (SDP) programming. W ...

2020

Recursive evaluation and iterative contraction of N-body equivariant features

Michele Ceriotti, Sergey Pozdnyakov, Jigyasa Nigam

Mapping an atomistic configuration to a symmetrized N-point correlation of a field associated with the atomic positions (e.g., an atomic density) has emerged as an elegant and effective solution to represent structures as the input of machine-learning algo ...

2020

Efficient Proximal Mapping of the 1-path-norm of Shallow Networks

Volkan Cevher, Paul Thierry Yves Rolland, Fabian Ricardo Latorre Gomez, Shaul Nadav Hallak

We demonstrate two new important properties of the 1-path-norm of shallow neural networks. First, despite its non-smoothness and non-convexity it allows a closed form proximal operator which can be efficiently computed, allowing the use of stochastic proxi ...

JMLR-JOURNAL MACHINE LEARNING RESEARCH2020

Stability of standing waves for a nonlinear Klein-Gordon equation with delta potentials

François Samer Genoud

In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point argument. Unlike ...

2019

Travelling heteroclinic waves in a Frenkel-Kontorova chain with anharmonic on-site potential

Boris Buffoni

The Frenkel-Kontorova model for dislocation dynamics from 1938 is given by a chain of atoms, where neighbouring atoms interact through a linear spring and are exposed to a smooth periodic on-site potential. A dislocation moving with constant speed correspo ...

ELSEVIER SCIENCE BV2019