arXiv subjects

Frédéric de Gournay

Publications and source records attributed to Frédéric de Gournay.

14 recordsLinked to original sources

arXiv2025

Irrationality as a mean of regularization in Bayesian Persuasion

We study a regularized variant of the Bayesian Persuasion problem, where the receiver's decision process includes a divergence-based penalty that accounts for deviations from perfect rationality. This modification smooths the underlying optimization landscape and mitigates key theoretical issues, such as measurability and ill-posedness, commonly encountered in the classical formulation. It also enables the use of scalable second-order optimization methods to compute numerically the optimal signaling scheme in a setting known to be NP-hard. We present theoretical results comparing the regularized and original models, including convergence guarantees and structural properties of optimal signaling schemes. Analytical examples and numerical simulations illustrate how this framework accommodates complex environments while remaining tractable and robust. A companion Python library, BASIL, makes use of all the practical insights from this article.

math.OC↗

arXiv2023

Bayesian Optimization of Sampling Densities in MRI

Data-driven optimization of sampling patterns in MRI has recently received a significant attention.Following recent observations on the combinatorial number of minimizers in off-the-grid optimization, we propose a framework to globally optimize the sampling densities using Bayesian optimization. Using a dimension reduction technique, we optimize the sampling trajectories more than 20 times faster than conventional off-the-grid methods, with a restricted number of training samples. This method -- among other benefits -- discards the need of automatic differentiation.Its performance is slightly worse than state-of-the-art learned trajectories since it reduces the space of admissible trajectories, but comes with significant computational advantages.Other contributions include: i) a careful evaluation of the distance in probability space to generate trajectories ii) a specific training procedure on families of operators for unrolled reconstruction networks and iii) a gradient projection based scheme for trajectory optimization.

eess.SP↗

arXiv2023

Grid is Good: Adaptive Refinement Algorithms for Off-the-Grid Total Variation Minimization

We propose an adaptive refinement algorithm to solve total variation regularized measure optimization problems. The method iteratively constructs dyadic partitions of the unit cube based on i) the resolution of discretized dual problems and ii) on the detection of cells containing points that violate the dual constraints. The detection is based on upper-bounds on the dual certificate, in the spirit of branch-and-bound methods. The interest of this approach is that it avoids the use of heuristic approaches to find the maximizers of dual certificates. We prove the convergence of this approach under mild hypotheses and a linear convergence rate under additional non-degeneracy assumptions. These results are confirmed by simple numerical experiments.

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arXiv2022

Adaptive scaling of the learning rate by second order automatic differentiation

In the context of the optimization of Deep Neural Networks, we propose to rescale the learning rate using a new technique of automatic differentiation. This technique relies on the computation of the {\em curvature}, a second order information whose computational complexity is in between the computation of the gradient and the one of the Hessian-vector product. If (1C,1M) represents respectively the computational time and memory footprint of the gradient method, the new technique increase the overall cost to either (1.5C,2M) or (2C,1M). This rescaling has the appealing characteristic of having a natural interpretation, it allows the practitioner to choose between exploration of the parameters set and convergence of the algorithm. The rescaling is adaptive, it depends on the data and on the direction of descent. The numerical experiments highlight the different exploration/convergence regimes.

cs.NE↗

arXiv2022

Spurious minimizers in non uniform Fourier sampling optimization

A recent trend in the signal/image processing literature is the optimization of Fourier sampling schemes for specific datasets of signals. In this paper, we explain why choosing optimal non Cartesian Fourier sampling patterns is a difficult nonconvex problem by bringing to light two optimization issues. The first one is the existence of a combinatorial number of spurious minimizers for a generic class of signals. The second one is a vanishing gradient effect for the high frequencies. We conclude the paper by showing how using large datasets can mitigate first effect and illustrate experimentally the benefits of using stochastic gradient algorithms with a variable metric.

math.OC↗

arXiv2022

CorticalFlow: A Diffeomorphic Mesh Deformation Module for Cortical Surface Reconstruction

In this paper we introduce CorticalFlow, a new geometric deep-learning model that, given a 3-dimensional image, learns to deform a reference template towards a targeted object. To conserve the template mesh's topological properties, we train our model over a set of diffeomorphic transformations. This new implementation of a flow Ordinary Differential Equation (ODE) framework benefits from a small GPU memory footprint, allowing the generation of surfaces with several hundred thousand vertices. To reduce topological errors introduced by its discrete resolution, we derive numeric conditions which improve the manifoldness of the predicted triangle mesh. To exhibit the utility of CorticalFlow, we demonstrate its performance for the challenging task of brain cortical surface reconstruction. In contrast to current state-of-the-art, CorticalFlow produces superior surfaces while reducing the computation time from nine and a half minutes to one second. More significantly, CorticalFlow enforces the generation of anatomically plausible surfaces; the absence of which has been a major impediment restricting the clinical relevance of such surface reconstruction methods.

cs.CV↗

arXiv2020

Off-the-grid data-driven optimization of sampling schemes in MRI

We propose a novel learning based algorithm to generate efficient and physically plausible sampling patterns in MRI. This method has a few advantages compared to recent learning based approaches: i) it works off-the-grid and ii) allows to handle arbitrary physical constraints. These two features allow for much more versatility in the sampling patterns that can take advantage of all the degrees of freedom offered by an MRI scanner. The method consists in a high dimensional optimization of a cost function defined implicitly by an algorithm. We propose various numerical tools to address this numerical challenge.

cs.LG↗

arXiv2019

Approximation of curves with piecewise constant or piecewise linear functions

In this paper we compute the Hausdorff distance between sets of continuous curves and sets of piecewise constant or linear discretizations. These sets are Sobolev balls given by the continuous or discrete $L^p$-norm of the derivatives. We detail the suitable discretization or smoothing procedure which are preservative in the sense of these norms. Finally we exhibit the link between Eulerian numbers and the uniformly space knots B-spline used for smoothing.

math.NA↗

arXiv2019

On the linear convergence rates of exchange and continuous methods for total variation minimization

We analyze an exchange algorithm for the numerical solution total-variation regularized inverse problems over the space M($Ω$) of Radon measures on a subset $Ω$ of R d. Our main result states that under some regularity conditions, the method eventually converges linearly. Additionally, we prove that continuously optimizing the amplitudes of positions of the target measure will succeed at a linear rate with a good initialization. Finally, we propose to combine the two approaches into an alternating method and discuss the comparative advantages of this approach.

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arXiv2019

Optimal Transport Approximation of 2-Dimensional Measures

We propose a fast and scalable algorithm to project a given density on a set of structured measures defined over a compact 2D domain. The measures can be discrete or supported on curves for instance. The proposed principle and algorithm are a natural generalization of previous results revolving around the generation of blue-noise point distributions, such as Lloyd's algorithm or more advanced techniques based on power diagrams. We analyze the convergence properties and propose new approaches to accelerate the generation of point distributions. We also design new algorithms to project curves onto spaces of curves with bounded length and curvature or speed and acceleration. We illustrate the algorithm's interest through applications in advanced sampling theory, non-photorealistic rendering and path planning.

math.NA↗

arXiv2018

Convex Regularization and Representer Theorems

We establish a result which states that regularizing an inverse problem with the gauge of a convex set $C$ yields solutions which are linear combinations of a few extreme points or elements of the extreme rays of $C$. These can be understood as the \textit{atoms} of the regularizer. We then explicit that general principle by using a few popular applications. In particular, we relate it to the common wisdom that total gradient variation minimization favors the reconstruction of piecewise constant images.

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arXiv2018

3/4-discrete optimal transport

This paper deals with the 3/4-discrete 2-Wasserstein optimal transport between two measures, where one is supported by a set of segment and the other one is supported by a set of Dirac masses. We select the most suitable optimization procedure that computes the optimal transport and provide numerical examples of approximation of cloud data by segments.

math.NA↗

arXiv2018

Differentiation and regularity of semi-discrete optimal transport with respect to the parameters of the discrete measure

This paper aims at determining under which conditions the semi-discrete optimal transport is twice differentiable with respect to the parameters of the discrete measure and exhibits numerical applications. The discussion focuses on minimal conditions on the background measure to ensure differentiability. We provide numerical illustrations in stippling and blue noise problems.

math.NA↗

arXiv2018

The case of Neumann, Robin and periodic lateral condition for the semi infinite generalized Graetz problem and applications

The Graetz problem is a convection-diffusion equation in a pipe invariant along a direction. The contribution of the present work is to propose a mathematical analysis of the Neumann, Robin and periodic boundary condition on the boundary of a semi-infinite pipe. The solution in the 3D space of the original problem is reduced to eigenproblems in the 2D section of the pipe. The set of solutions is described, its structure depends on the type of boundary condition and of the sign of the total flow of the fluid. This analysis is the cornerstone of numerical methods to solve Graetz problem in finite pipes, semi infinite pipes and exchangers of arbitrary cross section. Numerical test-cases illustrate the capabilities of these methods to provide solutions in various configurations.

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