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Pierre Bousquet

Publications and source records attributed to Pierre Bousquet.

At least 19 recordsLinked to original sources
arXiv2026

Born Discrete, Made Smooth: Variational Formulation of Shallow Neural Networks

Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics. In this work, we propose a paradigm shift by replacing the discrete training problem of shallow neural networks with a well-posed continuum variational surrogate. We identify a family of $\lambda$-convex functionals over parameter densities in weighted Sobolev spaces and prove that these variational problems are globally well-posed, stable, and exhibit unexpected almost $C^3$ regularity. Unlike existing Wasserstein-based or Mean-Field approaches, which often face limited regularity and discretization challenges, our formulation provides direct access to elliptic regularity and convex analysis. This allows us to prove that the optimal parameter density can be obtained by solving a single linear system, bypassing iterative optimization entirely. We establish explicit generalization error controls at a rate of $1/\alpha$ relative to the regularization parameter, and prove that finite-width networks of size $N$ achieve the continuum optimum at an $O(1/N)$ rate. This perspective bridges the gap between the Neural Tangent Kernel (NTK) and feature-learning regimes, providing a principled framework for understanding over-parameterization through the lens of variational calculus.

stat.ML↗
arXiv2025

Generic topological screening and approximation of Sobolev maps

This manuscript develops a framework for the strong approximation of Sobolev maps with values in compact manifolds, emphasizing the interplay between local and global topological properties. Building on topological concepts adapted to VMO maps, such as homotopy and the degree of continuous maps, it introduces and analyzes extendability properties, focusing on the notions of $\ell$-extendability and its generalization, $(\ell, e)$-extendability. We rely on Fuglede maps, providing a robust setting for handling compositions with Sobolev maps. Several constructions -- including opening, thickening, adaptive smoothing, and shrinking -- are carefully integrated into a unified approach that combines homotopical techniques with precise quantitative estimates. Our main results establish that a Sobolev map $u \in W^{k, p}$ defined on a compact manifold of dimension $m > kp$ can be approximated by smooth maps if and only if $u$ is $(\lfloor kp \rfloor, e)$-extendable with $e = m$. When $e < m$, the approximation can still be carried out using maps that are smooth except on structured singular sets of rank $m - e - 1$.

math.FA↗
arXiv2024

Discarding Lavrentiev's Gap in Non-autonomous and Non-Convex Variational Problems

We establish that the Lavrentiev gap between Sobolev and Lipschitz maps does not occur for a scalar variational problem of the form: \[ \textrm{to minimize} \qquad u \mapsto \int_\Omega f(x,u,\nabla u)\,dx \,, \] under a Dirichlet boundary condition. Here, \(\Omega\) is a bounded Lipschitz open set in \(\rn\), \(N\geq 1\) and the function $f$ is required to be measurable with respect to the spatial variable, continuous with respect to the second one, and convex with respect to the last variable. Under these assumptions alone, Lavrentiev gaps may occur, as illustrated by classical examples in the literature. We identify an additional natural condition on $f$ to discard such phenomena, that can be interpreted as a balance between the variations with respect to the first variable and the growth with respect to the last one. This unifies most of the structural assumptions that have been introduced so far to prevent the occurence of Lavrentiev gaps. Remarkably, typical assumptions that are usually imposed on $f$ in this setting are dropped here: we do not require $f$ to be bounded or convex with respect to the second variable, nor impose any condition of $\Delta_2$-kind with respect to the last variable.

math.AP↗
arXiv2022

Planetary Exploration Horizon 2061 Report, Chapter 4: From planetary exploration goals to technology requirements

This chapter reviews for each province and destination of the Solar System the representative space missions that will have to be designed and implemented by 2061 to address the six key science questions about the diversity, origins, workings and habitability of planetary systems (described in chapter 1) and to perform the critical observations that have been described in chapters 3 and partly 2. It derives from this set of future representative missions, some of which will have to be flown during the 2041-2061 period, the critical technologies and supporting infrastructures that will be needed to fly these challenging missions, thus laying the foundation for the description of technologies and infrastructures for the future of planetary exploration that is given in chapters 5 and 6, respectively.

astro-ph.IM↗
arXiv2022

Singular orthotropic functionals with nonstandard growth conditions

We pursue the study of a model convex functional with orthotropic structure and nonstandard growth conditions, this time focusing on the sub-quadratic case. We prove that bounded local minimizers are locally Lipschitz. No restriction on the ratio between the highest and the lowest growth rates are needed. The result holds also in presence of a non-autonomous lower order term, under sharp integrability assumptions. Finally, we prove higher differentiability of bounded local minimizers, as well.

math.AP↗
arXiv2019

Hardy-Sobolev inequalities for vector fields and canceling linear differential operators

Given a homogeneous k-th order differential operator $A (D)$ on $\mathbb{R}^n$ between two finite dimensional spaces, we establish the Hardy inequality $$\int_{\mathbb{R}^n} \frac{\lvert D^{k-1}u\rvert}{\lvert x \rvert} \,\mathrm{d} x \leq C \int_{\mathbb{R}^n} \lvert A(D)u\rvert $$ and the Sobolev inequality $$\lVert D^{k-n} u\rVert_{L^{\infty}(\mathbb{R}^n)}\leq C \int_{\mathbb{R}^n} \lvert A(D)u\rvert $$ when $A(D)$ is elliptic and satisfies a recently introduced cancellation property. We also study the necessity of these two conditions.

math.FA↗
arXiv2019

The equation div$u$+$\langle a, u \rangle=f$

We study the solutions $u$ to the equation $$ \begin{cases} \operatorname{div} u + \langle a , u \rangle = f & \textrm{ in } Ω,\\ u=0 & \textrm{ on } \partial Ω, \end{cases} $$ where $a$ and $f$ are given. We significantly improve the existence results of [Csató and Dacorogna, A Dirichlet problem involving the divergence operator, \textit{Ann. Inst. H. Poincaré Anal. Non Linéaire}, 33 (2016), 829--848], where this equation has been considered for the first time. In particular, we prove the existence of a solution under essentially sharp regularity assumptions on the coefficients. The condition that we require on the vector field $a$ is necessary and sufficient. Finally, our results cover the whole scales of Sobolev and Hölder spaces.

math.AP↗
arXiv2018

Approximation in higher-order Sobolev spaces and Hodge systems

Let $d\geq 2$ be an integer, $1\leq l\leq d-1$ and $φ$ be a differential $l$-form on ${\mathbb R}^d$ with $\dot{W}^{1,d}$ coefficients. It was proved by Bourgain and Brezis (\cite[Theorem 5]{MR2293957}) that there exists a differential $l$-form $ψ$ on ${\mathbb R}^d$ with coefficients in $L^{\infty}\cap \dot{W}^{1,d}$ such that $dφ=dψ$. Bourgain and Brezis also asked whether this result can be extended to differential forms with coefficients in the fractional Sobolev space $\dot{W}^{s,p}$ with $sp=d$. We give a positive answer to this question, in the more general context of Triebel-Lizorkin spaces, provided that $d-κ\leq l\leq d-1$, where $κ$ is the largest positive integer such that $κ<\min(p,d)$. The proof relies on an approximation result for functions in $\dot{W}^{s,p}$ by functions in $\dot{W}^{s,p}\cap L^{\infty}$, even though $\dot{W}^{s,p}$ does not embed into $L^{\infty}$ in this critical case.

math.CA↗
arXiv2018

On the Lipschitz character of orthotropic $p-$harmonic functions

We prove that local weak solutions of the orthotropic $p-$harmonic equation are locally Lipschitz, for every $p\ge 2$ and in every dimension. More generally, the result holds true for more degenerate equations with orthotropic structure, with right-hand sides in suitable Sobolev spaces.

math.AP↗
arXiv2017

Density of bounded maps in Sobolev spaces into complete manifolds

Given a complete noncompact Riemannian manifold $N^n$, we investigate whether the set of bounded Sobolev maps $(W^{1, p} \cap L^\infty) (Q^m; N^n)$ on the cube $Q^m$ is strongly dense in the Sobolev space $W^{1, p} (Q^m; N^n)$ for $1 \le p \le m$. The density always holds when $p$ is not an integer. When $p$ is an integer, the density can fail, and we prove that a quantitative trimming property is equivalent with the density. This new condition is ensured for example by a uniform Lipschitz geometry of $N^n$. As a byproduct, we give necessary and sufficient conditions for the strong density of the set of smooth maps $C^\infty (\overline{Q^m}; N^n)$ in $W^{1, p} (Q^m; N^n)$.

math.FA↗
arXiv2017

Weak approximation by bounded Sobolev maps with values into complete manifolds

We have recently introduced the trimming property for a complete Riemannian manifold $N^{n}$ as a necessary and sufficient condition for bounded maps to be strongly dense in $W^{1, p}(B^m; N^{n})$ when $p \in \{1, \dotsc, m\}$. We prove in this note that even under a weaker notion of approximation, namely the weak sequential convergence, the trimming property remains necessary for the approximation in terms of bounded maps. The argument involves the construction of a Sobolev map having infinitely many analytical singularities going to infinity.

math.FA↗
arXiv2015

Global Lipschitz continuity for minima of degenerate problems

We consider the problem of minimizing the Lagrangian $\int [F(\nabla u)+f\,u]$ among functions on $Ω\subset\mathbb{R}^N$ with given boundary datum $φ$. We prove Lipschitz regularity up to the boundary for solutions of this problem, provided $Ω$ is convex and $φ$ satisfies the bounded slope condition. The convex function $F$ is required to satisfy a qualified form of uniform convexity {\it only outside a ball} and no growth assumptions are made.

math.AP↗
arXiv2015

Strong density for higher order Sobolev spaces into compact manifolds

Given a compact manifold $N^n$, an integer $k \in \mathbb{N}_*$ and an exponent $1 \le p < \infty$, we prove that the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k, p}(Q^m; N^n)$ when the homotopy group $π_{\lfloor kp \rfloor}(N^n)$ of order $\lfloor kp \rfloor$ is trivial. We also prove the density of maps that are smooth except for a set of dimension $m - \lfloor kp \rfloor - 1$, without any restriction on the homotopy group of $N^n$

math.FA↗
arXiv2014

Lipschitz regularity for local minimizers of some widely degenerate problems

We consider local minimizers of the functional \[ \sum_{i=1}^N \int (|u_{x_i}|-δ_i)^p_+\, dx+\int f\, u\, dx, \] where $δ_1,\dots,δ_N\ge 0$ and $(\,\cdot\,)_+$ stands for the positive part. Under suitable assumptions on $f$, we prove that local minimizers are Lipschitz continuous functions if $N=2$ and $p\ge 2$, or if $N\ge 2$ and $p\ge 4$.

math.AP↗
arXiv2014

Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell$ simply connected manifolds when $s \ge 1$

Given a compact manifold $N^n \subset \mathbb{R}^ν$, $s \ge 1$ and $1 \le p < \infty$, we prove that the class of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s, p}(Q^m; N^n)$ when $N^n$ is $\lfloor sp \rfloor$ simply connected. For $sp$ integer, we prove weak density of smooth maps with values into $N^n$ when $N^n$ is $sp - 1$ simply connected. The proofs are based on the existence of a retraction of $\mathbb{R}^ν$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} \cap W^{1, sp}$.

math.FA↗