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Albert Clop

Publications and source records attributed to Albert Clop.

At least 19 recordsLinked to original sources
arXiv2026

When does fusing hand-crafted knowledge with learned representations pay? A cost-normalized benchmark of stacking, substitution, and interference

Fusing prior knowledge with data-driven learning is attractive where data is scarce, yet no controlled account says when it helps, is redundant, or harms. We benchmark one fixed hand-crafted knowledge source, a pinned bank of Gabor targets injected only during training at $\sim$2\% overhead, against data-driven alternatives (SimCLR, SimSiam, DINO, ImageNet transfer, augmentation, learned teachers) under one frozen recipe with fixed subsets: 13 datasets, 9 backbones, 150 to 1.28M images, 32--224\,px, 2.5M--86M parameters ($\computeCells$ classification configurations over $\computeRuns$ runs, plus segmentation and detection transplants). Across the training-time combinations we measure, three outcomes recur (decision-level fusion differs). Different-\emph{currency} sources can stack: the prior composes with DeiT augmentation on attention backbones and is worth $+26$ points to ViT-B/16 at $224$\,px, $+6.7$ at twice that budget. Same-currency sources substitute: against effective self-supervised pretraining, the combination never usefully exceeds the better single source. Fusing at full strength into an already-informed initialization interferes in proportion to what it carries: ImageNet transfer, $-15$ to $-17$ points, removed by a weaker auxiliary weight. Frozen-feature diagnostics measured on each source alone separate these outcomes retrospectively but do not predict them: a rule built on them calls one of nine unseen pairs. At a practitioner's own label budget, the frozen-feature gain predicts the end-to-end gain to within $0.17$ points across 30 cells and seven datasets; the underlying decomposition, $\Delta = G + \readout(\mathrm{base})$, holds in sign on $\auditRate\%$ of testable cells and is called an unseen backbone family's feature gain in advance. The project page is https://amughrabi.github.io/MomentAux.

cs.CV↗
arXiv2026

Regularity for Minimizers of non Autonomous Singular Functionals with Anisotropic Growth

We establish the higher differentiability of the local minimizers to a class of non autonomous convex integral functionals satisfying anisotropic subquadratic growth conditions, that include, as a particular case, those with orthotropic structure. The result is obtained under a gap bound on the exponents \(p_i\), that guarantees the local boundedness of the minimizers and under a suitable Sobolev assumption on the map that measures the oscillation of the energy density with respect to the $x$ variable, that is independent on the dimension.

math.AP↗
arXiv2024

Lipschitz regularity for solutions to an orthotropic $q$-Laplacian-type equation in the Heisenberg group

We establish the local Lipschitz regularity for solutions to an orthotropic q-Laplacian-type equation within the Heisenberg group. Our approach is largely inspired by the works of X. Zhong, who investigated the q-Laplacian in the same setting and proved the H\"older regularity for the gradient of solutions. Due to the degeneracy of the current equation, such regularity for the gradient of solutions is not even known in the Euclidean setting for dimensions greater than 2, where only boundedness is expected.

math.AP↗
arXiv2024

A continuous model of transportation in the Heisenberg group

We present a minimization problem with a horizontal divergence-type constraint in the Heisenberg group. Our study explores its dual formulation and examines its relationship with the congested optimal transport problem, for $1 < p < +\infty$, as well as the Monge-Kantorovich problem, in the limite case $p=1$.

math.AP↗
arXiv2022

Nonlinear transport equations and quasiconformal maps

We prove existence of solutions to a nonlinear transport equation in the plane, for which the velocity field is obtained as the convolution of the classical Cauchy Kernel with the unknown. Even though the initial datum is bounded and compactly supported, the velocity field may have unbounded divergence. The proof is based on the compactness property of quasiconformal mappings.

math.AP↗
arXiv2022

Higher differentiability results for solutions to a class of non-homogeneouns elliptic problems under sub-quadratic growth conditions

We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,\Omega\right)=\int_{\Omega}\left[ F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand $F(x,\xi)$ which is convex with respect to the gradient variable, under $p$-growth conditions, with $1<p<2$. The main novelty here is that the results are obtained assuming that the partial map $x\mapsto D_\xi F(x,\xi)$ has weak derivatives in some Lebesgue space $L^q$ and the datum $f$ is assumed to belong to a suitable Lebesgue space $L^r$. We also prove that it is possible to weaken the assumption on the datum $f$ and on the map $x\mapsto D_\xi F(x,\xi)$, if the minimizers are assumed to be a priori bounded.

math.AP↗
arXiv2021

Rotation bounds for H\"older continuous homeomorphisms with integrable distortion

We obtain sharp rotation bounds for the subclass of homeomorphisms $f:\mathbb{C}\to\mathbb{C}$ of finite distortion which have distortion function in $L^p_{loc}$, $p>1$, and for which a H\"older continuous inverse is available. The interest in this class is partially motivated by examples arising from fluid mechanics. Our rotation bounds hereby presented improve the existing ones, for which the H\"older continuity is not assumed. We also present examples proving sharpness.

math.AP↗
arXiv2020

Pointwise descriptions of nearly incompressible vector fields with bounded curl

Among those nearly incompressible vector fields ${\bf{v}}:{\mathbb{R}}^n\to{\mathbb{R}}^n$ with $|x|\log|x|$ growth at infinity, we give a pointwise characterization of the ones for which $\operatorname{curl}{\bf{v}}= D{\bf{v}}-D^t{\bf{v}}$ belongs to $L^\infty$. When $n=2$ we can go further and describe, still in pointwise terms, the vector fields ${\bf{v}}:{\mathbb{R}}^2\to{\mathbb{R}}^2$ for which $|\operatorname{div}{\bf{v}}|+|\operatorname{curl}{\bf{v}}|\in L^\infty$.

math.CA↗
arXiv2019

Improved H\"older regularity for strongly elliptic PDEs

We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new relations between these two equations. In particular, we show that solutions to an autonomous Beltrami equation enjoy a quantitative improved degree of H\"older regularity, higher than what is given by the classical exponent $1/K$.

math.CV↗
arXiv2018

Very degenerate elliptic equations under almost critical Sobolev regularity

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of integral functionals with non autonomous integrand which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the coefficients have weak derivative in an almost critical Zygmund class and the datum f is assumed to belong to the same Zygmund class.

math.AP↗
arXiv2016

Fractional differentiability for solutions of nonlinear elliptic equations

We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,ξ)$ enjoys $B^α_{\frac{n}α, q}$ smoothness, local well-posedness is found in $B^α_{p,q}$ for certain values of $p\in[2,\frac{n}α)$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,ξ)=A(x)ξ$, $G=0$ and $A\in B^α_{\frac{n}α,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^α_{p,q}$ for each $p<\frac{n}α$. Our main tool in the proof is a more general result, that holds also if ${\mathcal A}$ has growth $s-1$ in $Du$, $2\leq s\leq n$, and asserts local well-posedness in $L^q$ for each $q>s$, provided that $x\mapsto{\mathcal A}(x,ξ)$ satisfies a locally uniform $VMO$ condition.

math.AP↗
arXiv2016

Flows for non-smooth vector fields with subexponentially integrable divergence

In this paper, we study flows associated to Sobolev vector fields with subexponentially integrable divergence. Our approach is based on the transport equation following DiPerna-Lions [DPL89]. A key ingredient is to use a quantitative estimate of solutions to the Cauchy problem of transport equation to obtain the regularity of density functions.

math.CA↗
arXiv2015

Beltrami equations with coefficient in the fractional Sobolev space $W^{θ, \frac2θ}$

In this paper, we look at quasiconformal solutions $ϕ:\mathbb{C}\to\mathbb{C}$ of Beltrami equations $$ \partial_{\overline{z}} ϕ(z)=μ(z)\,\partial_z ϕ(z). $$ where $μ\in L^\infty(\mathbb{C})$ is compactly supported on $\mathbb{D}$, $\|μ\|_\infty<1$ and belongs to the fractional Sobolev space $W^{α, \frac2α}(\mathbb{C})$. Our main result states that $$\log\partial_zϕ\in W^{α, \frac2α}(\mathbb{C})$$ whenever $α>\frac12$. Our method relies on an $n$-dimensional result, which asserts the compactness of the commutator $$[b,(-Δ)^\fracβ{2}]:L^\frac{np}{n-βp}(\mathbb{R}^n)\to L^p(\mathbb{R}^n)$$ between the fractional laplacian $(-Δ)^\frac\beta2$ and any symbol $b\in W^{β,\frac{n}β}(\mathbb{R}^n)$, provided that $1<p<\frac{n}β$.

math.CV↗
arXiv2015

Linear transport equations for vector fields with subexponentially integrable divergence

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline integrability assumptions on the divergence of the velocity field $b$. For $W^{1,1}_{loc}$ vector fields $b$ satisfying $\frac{|b(x,t)|}{1+|x|}\in L^1(0,T; L^1)+L^1(0,T; L^\infty)$ and $$\operatorname{div} b\in L^1(0,T;L^\infty) + L^1\left(0,T; \operatorname{Exp}\left(\frac{L}{\log L}\right)\right),$$ we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every $γ>1$, we construct an example of a bounded autonomous velocity field $b$ with $$\operatorname{div} b\in \operatorname{Exp}\left(\frac{L}{\log^γL}\right) ,$$ for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the $BV$ setting are also addressed.

math.AP↗
arXiv2014

Manifolds of quasiconformal mappings and the nonlinear Beltrami equation

In this paper we show that the homeomorphic solutions to each nonlinear Beltrami equation $\partial_{\bar{z}} f = \mathcal{H}(z, \partial_{z} f)$ generate a two-dimensional manifold of quasiconformal mappings $\mathcal{F}_{\mathcal{H}} \subset W^{1,2}_{\mathrm{loc}}(\mathbb{C})$. Moreover, we show that under regularity assumptions on $\mathcal{H}$, the manifold $\mathcal{F}_{\mathcal{H}}$ defines the structure function $\mathcal{H}$ uniquely.

math.CV↗