arXiv subjects

Roberto Colombo

Publications and source records attributed to Roberto Colombo.

8 recordsLinked to original sources

arXiv2026

The fractional Laplacian in Lipschitz domains: Dahlberg's Theorem and $L^{2}$-solvability

Given $s\in (0,1)$ and a bounded Lipschitz domain $\Omega\subset \mathbb{R}^{n}$, we establish a quantitative Dahlberg theory for the $s$-harmonic measure of $\Omega$, $\omega_s^x$. In the nonlocal setting, the natural reference measure is an integral weight $\sigma_{s}$ in $\Omega^{c}$ that behaves like $(1-s)\text{dist}(\cdot, \partial \Omega)^{-s}$ close to the boundary. Our main result is a scale-invariant reverse-H\"{o}lder estimate for the density $d\omega_{s}^{x}/d\sigma_{s}$ on boundary-centered balls. As a consequence, we obtain $L^2(\Omega^c,\sigma_s)$-solvability of the exterior Dirichlet problem, with estimates for a nonlocal non-tangential maximal function and uniqueness in the natural distributional class. A weighted Gehring argument improves the reverse-H\"{o}lder exponent beyond $2$ and consequently yields $L^{q}$-solvability for a range of exponents extending strictly below $2$. Our results apply to general symmetric stable operators comparable to the fractional Laplacian. Moreover, the proofs are compatible with the limit $s\to 1^-$ and thus yield the corresponding results for the Laplacian in the nonlocal-to-local limit. The main new step is to convert a fractional Pohozaev identity for the Green function into uniform square estimates on distance level sets of a Lipschitz boundary. As applications, we derive optimal Sobolev regularity estimates for the homogeneous weighted Dirichlet problem and for the inhomogeneous Poisson problem with zero exterior data.

math.AP↗

arXiv2026

Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow

Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last iterate. In this paper, we establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the $d$-dimensional torus. Specifically, assuming that the initial density and the target are smooth and close in $L^2$-norm, we obtain explicit polynomial convergence rates in $L^2$-norm that depend on the dimension and on the regularity parameters of the kernel, the initialization and the target. We further show that these rates are sharp in certain regimes, and support the theory with numerical experiments. In the edge case of kernels with a Coulomb singularity, we recover the global exponential convergence result established in prior work. Our analysis is inspired by recent results on Wasserstein gradient flows of kernel mean discrepancies.

stat.ML↗

arXiv2026

Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies

We study the quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancy (KMD) (also known as Maximum Mean Discrepancy (MMD)) functionals. Our setting covers in particular the training dynamics of shallow neural networks in the infinite-width and continuous time limit, as well as interacting particle systems with pairwise Riesz kernel interaction in the mean-field and overdamped limit. Our main analysis concerns the model case of KMD functionals given by the squared Sobolev distance $ \mathscr{E}^{\nu}_{s}(\mu)= \frac{1}{2}\lVert \mu-\nu\rVert_{\dot H^{-s}}^{2}$ for any $s\geq 1 $ and $\nu$ a fixed probability measure on the $d$-dimensional torus. First, inspired by Yudovich theory for the $2d$-Euler equation, we establish existence and uniqueness in natural weak regularity classes. Next, we show that for $s=1$ the flow converges globally at an exponential rate under minimal assumptions, while for $s>1$ we prove local convergence at polynomial rates that depend explicitly on $s$ and on the Sobolev regularity of $\mu$ and $\nu$. These rates hold both at the energy level and in higher regularity classes and are tight for $\nu$ uniform. We then consider the gradient flow of the population loss for shallow neural networks with ReLU activation, which can be cast as a Wasserstein--Fisher--Rao gradient flow on the space of nonnegative measures on the sphere $\mathbb{S}^d$. Exploiting a correspondence with the Sobolev energy case with $s=(d+3)/2$, we derive an explicit polynomial local convergence rate for this dynamics. Except for the special case $s=1$, even non-quantitative convergence was previously open in all these settings. We also include numerical experiments in dimension $d=1$ using both PDE and particle methods which illustrate our analysis.

math.AP↗

arXiv2026

Sharpness of the Osgood Criterion for the Continuity Equation with Divergence-free Vector Fields

For any modulus of continuity $\omega$ that fails the Osgood condition, we construct a divergence-free velocity field $v \in C_t C^\omega_x$ for which the associated ODE admits at least two distinct flow maps. In other words, non-uniqueness does not occur merely for a single or even finitely many trajectories, but instead on a set of initial conditions $E$ of positive Lebesgue measure. In fact, the set $E$ has full measure inside a cube where the construction is supported. Moreover, we also construct a divergence-free velocity field $v \in C_{t}C^\omega_x$ for which the associated continuity equation admits two distinct solutions $\mu^1$ and $\mu^2$ which are absolutely continuous with respect to Lebesgue measure for almost every time, and start from the same initial datum $\bar \mu \ll \mathscr{L}^{d}$. Our construction introduces two novel ideas: (i) We introduce the notion of "parallelization", where at each time, the velocity field consists of simultaneous motion across multiple nested spatial scales. This differs from most explicit constructions in the literature on mixing or anomalous dissipation, where the velocity on different scales acts at separate times. This is crucial to cover the whole class of non-Osgood moduli of continuity. (ii) Inspired by a recent work of Bru\`e, Colombo and Kumar, we develop a new fixed-point framework that naturally incorporates the parallelization mechanism. This framework allows us to construct anomalous solutions of the continuity equation that belong to $L^1(\mathbb{R}^d)$ a.e. in time.

math.AP↗

arXiv2025

A convex integration scheme for the continuity equation past the Sobolev embedding threshold

We introduce a convex integration scheme for the continuity equation in the context of the Di Perna-Lions theory that allows to build incompressible vector fields in $C_{t}W^{1,p}_x$ and nonunique solutions in $C_{t} L^{q}_x$ for any $p,q$ with $\frac{1}{p} + \frac{1}{q} > 1 + \frac{1}{d}- \delta$ for some $\delta>0$. This improves the previous bound, corresponding to $\delta=0$, or equivalently $q' > p^*$, obtained with convex integration so far, and critical for those schemes in view of the Sobolev embedding that guarantees that solutions are distributional in the opposite range.

math.AP↗

arXiv2024

Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian

We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian $(-\Delta)^s$. We prove that, for almost every obstacle, the free boundary contains only regular points up to dimension $3$, for every $s\in(0,1)$. To do so, we extend some results on the fine structure of the free boundary to the case $s\in (0,1)$ and general non-zero obstacle, including a blow-up analysis at points with frequency $2m+2s$, and we prove new explicit uniform frequency gaps for solutions of the fractional obstacle problem.

math.AP↗

arXiv2023

Partial regularity for minimizers of a class of discontinuous Lagrangians

We study a one dimensional Lagrangian problem including the variational reformulation, derived in a recent work of Ambrosio-Baradat-Brenier, of the discrete Monge-Amp\`ere gravitational model, which describes the motion of interacting particles whose dynamics is ruled by the optimal transport problem. The more general action type functional we consider contains a discontinuous potential term related to the descending slope of the opposite squared distance function from a generic discrete set in R^{d}. We exploit the underlying geometrical structure provided by the associated Voronoi decomposition of the space to obtain C^{1,1} regularity for local minimizers out of a finite number of shock times.

math.AP↗

arXiv2022

A deep scalable neural architecture for soil properties estimation from spectral information

In this paper we propose an adaptive deep neural architecture for the prediction of multiple soil characteristics from the analysis of hyperspectral signatures. The proposed method overcomes the limitations of previous methods in the state of art: (i) it allows to predict multiple soil variables at once; (ii) it permits to backtrace the spectral bands that most contribute to the estimation of a given variable; (iii) it is based on a flexible neural architecture capable of automatically adapting to the spectral library under analysis. The proposed architecture is experimented on LUCAS, a large laboratory dataset and on a dataset achieved by simulating PRISMA hyperspectral sensor. 'Results, compared with other state-of-the-art methods confirm the effectiveness of the proposed solution.

cs.CV↗