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Diego Pallara

Publications and source records attributed to Diego Pallara.

At least 19 recordsLinked to original sources
arXiv2026

The infinite volume limit of the Hopfield neural networks in the replica-symmetric regime

Despite extensive research over the past five decades, a rigorous determination of the infinite-volume limit of the free energy of the Hopfield model with an extensive number of patterns has remained an outstanding open problem as standard methods from spin-glass theory have so far proven inadequate for this purpose. While a complete proof for finite, non-vanishing, loads is still out of reach, here we develop a novel interpolation strategy that allows us to express the free energy of the Hopfield model in terms of a linear combination of the free energies of hard- and soft-spin glass models along with a remainder contribution. The payoff of this interpolation scheme is twofold: (i) As the remainder involves positive-definite fluctuations of the order parameters (and it vanishes under a concentration assumption on their means where we consistently recover the replica-symmetric picture) such an expression finally constitutes a rigorous {\em bound} on the exact free energy, rather than just an {\em approximation,} as it was the case for earlier interpolation procedures à la Guerra. \newline (ii) As for the spin-glass models the existence of the infinite-volume free energy has been previously determined by the Guerra--Toninelli scheme, this novel interpolation allows us to extend its validity to the Hopfield model too. Yet, as the equivalence between the Hopfield model and mixtures of spin-glasses is currently well established under replica symmetry and up to the first step of replica symmetry breaking, at present our approach can be rigorously implemented only within these regimes. \newline In this work, by inspecting how the order parameters have to self-average around their means, we restrict our attention to the replica-symmetric regime.

cond-mat.dis-nn↗
arXiv2026

On the optimization of the Robin eigenvalues in some classes of polygons

Given the eigenvalue problem for the Laplacian with Robin boundary conditions, (with $β\in\R\setminus\{0\}$ the Robin parameter), we consider shape minimization problems if $β>0$ and shape maximization problems if $β<0$. Both problems are settled in a suitable class of generalized polygons with an upper bound on the number of sides, under either perimeter or volume constraint.

math.AP↗
arXiv2024

Local Regularity of very weak $s$-harmonic functions via fractional difference quotients

The aim of this paper is to give a new proof that any very weak $s$-harmonic function $u$ in the unit ball $B$ is smooth. As a first step, we improve the local summability properties of $u$. Then, we exploit a suitable version of the difference quotient method tailored to get rid of the singularity of the integral kernel and gain Sobolev regularity and local linear estimates of the $H^{s}_{\rm loc}$ norm of $u$. Finally, by applying more standard methods, such as elliptic regularity and Schauder estimates, we reach real analyticity of $u$. Up to the authors' knowledge, the difference quotient techniques are new.

math.AP↗
arXiv2024

Hebbian Learning from First Principles

Recently, the original storage prescription for the Hopfield model of neural networks -- as well as for its dense generalizations -- has been turned into a genuine Hebbian learning rule by postulating the expression of its Hamiltonian for both the supervised and unsupervised protocols. In these notes, first, we obtain these explicit expressions by relying upon maximum entropy extremization à la Jaynes. Beyond providing a formal derivation of these recipes for Hebbian learning, this construction also highlights how Lagrangian constraints within entropy extremization force network's outcomes on neural correlations: these try to mimic the empirical counterparts hidden in the datasets provided to the network for its training and, the denser the network, the longer the correlations that it is able to capture. Next, we prove that, in the big data limit, whatever the presence of a teacher (or its lacking), not only these Hebbian learning rules converge to the original storage prescription of the Hopfield model but also their related free energies (and, thus, the statistical mechanical picture provided by Amit, Gutfreund and Sompolinsky is fully recovered). As a sideline, we show mathematical equivalence among standard Cost functions (Hamiltonian), preferred in Statistical Mechanical jargon, and quadratic Loss Functions, preferred in Machine Learning terminology. Remarks on the exponential Hopfield model (as the limit of dense networks with diverging density) and semi-supervised protocols are also provided.

cond-mat.dis-nn↗
arXiv2024

Stability of the Gaussian Faber-Krahn inequality

We prove a quantitative version of the Gaussian Faber-Krahn type inequality proved by Betta, Chiacchio and Ferone for the first Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator, estimating the deficit in terms of the Gaussian Fraenkel asymmetry. As expected, the multiplicative constant only depends on the prescribed Gaussian measure.

math.AP↗
arXiv2023

Mean value formulas for classical solutions to subelliptic evolution equations in stratified Lie groups

We prove mean value formulas for classical solutions to second order linear differential equations in the form $$ \partial_t u = \sum_{i,j=1}^m X_i (a_{ij} X_j u) + X_0 u + f, $$ where $A = (a_{ij})_{i,j=1, \dots,m}$ is a bounded, symmetric and uniformly positive matrix with $C^1$ coefficients under the assumption that the operator $\sum_{j=1}^m X_j^2 + X_0 - \partial_t$ is hypoelliptic and the vector fields $X_1, \dots, X_m$ and $X_{m+1} :=X_0 - \partial_t$ are invariant with respect to a suitable homogeneous Lie group. Our results apply e.g. to degenerate Kolmogorov operators and parabolic equations on Carnot groups $\partial_t u = \sum_{i,j=1}^m X_i (a_{ij} X_j u) + f$.

math.AP↗
arXiv2022

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relevant the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeters on stratified Lie groups.

math.AP↗
arXiv2021

Mean value formulas for classical solutions to uniformly parabolic equations in divergence form

We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in divergence form. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C^1 boundary, the other stated for sets with finite perimeter.

math.AP↗
arXiv2021

Gamma-convergence of fractional Gaussian perimeter

We prove the $Γ$-convergence of the renormalised fractional Gaussian $s$-perimeter to the Gaussian perimeter as $s\to 1^-$. Our definition of fractional perimeter comes from that of the fractional powers of Ornstein-Uhlenbeck operator given via Bochner subordination formula. As a typical feature of the Gaussian setting, the constant appearing in front of the $Γ$-limit does not depend on the dimension.

math.AP↗
arXiv2021

Asymptotics of the $s$-fractional Gaussian perimeter as $s\to 0^+$

We study the asymptotic behaviour of the renormalised $s$-fractional Gaussian perimeter of a set $E$ inside a domain $Ω$ as $s\to 0^+$. Contrary to the Euclidean case, as the Gaussian measure is finite, the shape of the set at infinity does not matter, but, surprisingly, the limit set function is never additive.

math.AP↗
arXiv2020

Ornstein-Uhlenbeck semigroups in infinite dimension

This is a survey paper about Ornstein-Uhlenbeck semigroups in infinite dimension, and their generators. We start from the classical Ornstein-Uhlenbeck semigroup in Wiener spaces and then discuss the general case in Hilbert spaces. Finally, we present some results for O-U semigroups in Banach spaces.

math.AP↗
arXiv2020

Local minimizers and Gamma-convergence for nonlocal perimeters in Carnot Groups

We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi's rectifiability Theorem holds, we provide a lower bound for the $Γ$-liminf of the rescaled energy in terms of the horizontal perimeter.

math.AP↗
arXiv2018

Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite dimensional convex domains

Let $X$ be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure $γ$ and let $λ_1$ be the maximum eigenvalue of the covariance operator associated with $γ$. The associated Cameron--Martin space is denoted by $H$. For a sufficiently regular convex function $U:X\to\mathbb{R}$ and a convex set $Ω\subseteq X$, we set $ν:=e^{-U}γ$ and we consider the semigroup $(T_Ω(t))_{t\geq 0}$ generated by the self-adjoint operator defined via the quadratic form \[ (φ,ψ)\mapsto \int_Ω\langle D_Hφ,D_Hψ\rangle_Hdν, \] where $φ,ψ$ belong to $D^{1,2}(Ω,ν)$, the Sobolev space defined as the domain of the closure in $L^2(Ω,ν)$ of $D_H$, the gradient operator along the directions of $H$. A suitable approximation procedure allows us to prove some pointwise gradient estimates for $(T_Ω(t))_{t\ge 0}$. In particular, we show that \[ |D_H T_Ω(t)f|_H^p\le e^{- p λ_1^{-1} t}(T_Ω(t)|D_H f|^p_H), \qquad\, t>0,\ ν\textrm{ -a.e. in }Ω, \] for any $p\in [1,+\infty)$ and $f\in D^{1,p}(Ω,ν)$. We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in $Ω$ for the measure $ν$ and some improving summability properties for $(T_Ω(t))_{t\geq 0}$. In addition we prove that if $f$ belongs to $L^p(Ω,ν)$ for some $p\in(1,\infty)$, then \[|D_H T_Ω(t)f|^p_H \leq K_p t^{-\frac{p}{2}} T_Ω(t)|f|^p,\qquad \, t>0,\ ν\text{-a.e. in }Ω,\] where $K_p$ is a positive constant depending only on $p$. Finally we investigate on the asymptotic behaviour of the semigroup $(T_Ω(t))_{t\geq 0}$ as $t$ goes to infinity.

math.AP↗
arXiv2016

The ground state of long-range Schrodinger equations and static $q\bar{q}$ potential

Motivated by the recent results in arXiv:1601.05679 about the quark-antiquark potential in $\mathcal N=4$ SYM, we reconsider the problem of computing the asymptotic weak-coupling expansion of the ground state energy of a certain class of 1d Schrödinger operators $-\frac{d^{2}}{dx^{2}}+λ\,V(x)$ with long-range potential $V(x)$. In particular, we consider even potentials obeying $\int_{\mathbb R}dx\, V(x)<0$ with large $x$ asymptotics $V\sim -a/x^{2}-b/x^{3}+\cdots$. The associated Schrödinger operator is known to admit a bound state for $λ\to 0^{+}$, but the binding energy is rigorously non-analytic at $λ=0$. Its asymptotic expansion starts at order $\mathcal O(λ)$, but contains higher corrections $λ^{n}\,\log^{m}λ$ with all $0\le m\le n-1$ and standard Rayleigh-Schrödinger perturbation theory fails order by order in $λ$. We discuss various analytical tools to tame this problem and provide the general expansion of the binding energy at $\mathcal O(λ^{3})$ in terms of quadratures. The method is tested on a soluble potential that is fully under control, and on various non-soluble cases as well. A supersymmetric case, arising in the study of the quark-antiquark potential in $\mathcal N=6$ ABJ(M) theory, is also exploited to provide a further non-trivial consistency check. Our analytical results confirm at third order a remarkable exponentiation of the leading infrared logarithms, first noticed in $\mathcal N=4$ SYM where it may be proved by Renormalization Group arguments. We prove this interesting feature at all orders at the level of the Schrödinger equation for general potentials in the considered class.

hep-th↗
arXiv2015

$L^p$-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space $L^p({\mathbb R}^d;{\mathbb R}^m)$, $(d,m \ge 1)$ with $p\in [1,+\infty)$. Sufficient conditions for the associated evolution operator ${\bf G}(t,s)$ in $C_b({\mathbb R}^d;{\mathbb R}^m)$ to extend to a strongly continuous operator in $L^p({\mathbb R}^d;{\mathbb R}^m)$ are given. Some $L^p$-$L^q$ estimates are also established together with $L^p$ gradient estimates.

math.AP↗
arXiv2014

$\mathsf{L}^1$-elliptic regularity and $H=W$ on the whole $\mathsf{L}^p$-scale on arbitrary manifolds

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial differential operators, and we prove the density of the corresponding smooth Sobolev sections in these spaces under a generalized ellipticity condition on the underlying family. In particular, this implies a covariant version of Meyers-Serrin\rq{}s theorem on the whole $\mathsf{L}^p$-scale, for arbitrary Riemannian manifolds. Furthermore, we prove a new local elliptic regularity result in $\mathsf{L}^1$ on the Besov scale, which shows that the above generalized ellipticity condition is satisfied on the whole $\mathsf{L}^p$-scale, if some differential operator from $\mathfrak{P}$ that has a sufficiently high (but not necessarily the highest) order is elliptic.

math.AP↗