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Gian Paolo Leonardi

Publications and source records attributed to Gian Paolo Leonardi.

At least 19 recordsLinked to original sources
arXiv2026

Convex sets with vanishing relative width and the reverse Cheeger inequality

We study the reverse Cheeger inequality, which bounds from above the ratio of the first Dirichlet Laplacian eigenvalue to the square of the Cheeger constant. We first extend this inequality from convex sets to a broader class. We then analyze maximizing sequences among convex bodies. To quantify domain collapse, we introduce the notion of principal widths. For three-dimensional convex bodies, we prove that if the ratio of the first principal width to the second principal width vanishes along a sequence of convex bodies, then such sequence is maximizing. Finally, in arbitrary dimensions, we prove that the same property holds for the class of rhomboid-like sets.

math.SP↗
arXiv2025

Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flow

We propose a construction of mean curvature flows by approximation for very general initial data, in the spirit of the works of Brakke and of Kim & Tonegawa based on the theory of varifolds. Given a general varifold, we construct by iterated push-forwards an approximate time-discrete mean curvature flow depending on both a given time step and an approximation parameter. We show that, as the time step tends to $0$, this time-discrete flow converges to a unique limit flow, which we call the approximate mean curvature flow. An interesting feature of our approach is its generality, as it provides an approximate notion of mean curvature flow for very general structures of any dimension and codimension, ranging from continuous surfaces to discrete point clouds. We prove that our approximate mean curvature flow satisfies several properties: stability, uniqueness, Brakke-type equality, mass decay. By coupling this approximate flow with the canonical time measure, we prove convergence, as the approximation parameter tends to $0$, to a spacetime limit measure whose generalized mean curvature is bounded. Under an additional rectifiability assumption, we further prove that this limit measure is a spacetime Brakke flow.

math.DG↗
arXiv2025

Stability of axial free-boundary hyperplanes in circular cones

Given an axially-symmetric, $(n+1)$-dimensional convex cone $Ω\subset \mathbb{R}^{n+1}$, we study the stability of the free-boundary minimal surface $Σ$ obtained by intersecting $Ω$ with a $n$-plane that contains the axis of $Ω$. In the case $n=2$, $Σ$ is always unstable, as a special case of the vertex-skipping property that we recently proved in another article. Conversely, as soon as $n \ge 3$ and $Ω$ has a sufficiently large aperture (depending on the dimension $n$), we show that $Σ$ is strictly stable. For our stability analysis, we introduce a Lipschitz flow $Σ_{t}[f]$ of deformations of $Σ$ associated with a compactly-supported, scalar deformation field $f$, which satisfies the key property $\partial Σ_{t}[f] \subset \partial Ω$ for all $t\in \mathbb{R}$. Then, we compute the lower-right second variation of the area of $Σ$ along the flow, and ultimately show that it is positive by exploiting its connection with a functional inequality studied in the context of reaction-diffusion problems.

math.AP↗
arXiv2025

Free-Boundary Monotonicity for Almost-Minimizers of the Relative Perimeter

Let $E \subset Ω$ be a local almost-minimizer of the relative perimeter in the open set $Ω\subset \mathbb{R}^{n}$. We prove a free-boundary monotonicity inequality for $E$ at a point $x\in \partialΩ$, under a geometric property called ``visibility'', that $Ω$ is required to satisfy in a neighborhood of $x$. Incidentally, the visibility property is satisfied by a considerably large class of Lipschitz and possibly non-smooth domains. Then, we prove the existence of the density of the relative perimeter of $E$ at $x$, as well as the fact that any blow-up of $E$ at $x$ is necessarily a perimeter-minimizing cone within the tangent cone to $Ω$ at $x$.

math.CA↗
arXiv2025

The Quantitative Faber-Krahn Inequality for the Combinatorial Laplacian in $\mathbb{Z}^{d}$

While the classical Faber-Krahn inequality shows that the ball uniquely minimizes the first Dirichlet eigenvalue of the Laplacian in the continuum, this rigidity may fail in the discrete setting. We establish quantitative fluctuation estimates for the first Dirichlet eigenvalue of the combinatorial Laplacian on subsets of $\mathbb{Z}^{d}$ when their cardinality diverges. Our approach is based on a controlled discrete-to-continuum extension of the associated variational problem and the quantitative Faber-Krahn inequality.

math.FA↗
arXiv2025

A Two-Scale Complexity Measure for Deep Learning Models

We introduce a novel capacity measure 2sED for statistical models based on the effective dimension. The new quantity provably bounds the generalization error under mild assumptions on the model. Furthermore, simulations on standard data sets and popular model architectures show that 2sED correlates well with the training error. For Markovian models, we show how to efficiently approximate 2sED from below through a layerwise iterative approach, which allows us to tackle deep learning models with a large number of parameters. Simulation results suggest that the approximation is good for different prominent models and data sets.

stat.ML↗
arXiv2025

Measures in the dual of $BV$: perimeter bounds and relations with divergence-measure fields

We analyze some properties of the measures in the dual of the space $BV$, by considering (signed) Radon measures satisfying a perimeter bound condition, which means that the absolute value of the measure of a set is controlled by the perimeter of the set itself, and whose total variations also belong to the dual of $BV$. We exploit and refine the results of [25](Phuc, Torres 2017), in particular exploring the relation with divergence-measure fields and proving the stability of the perimeter bound from sets to $BV$ functions under a suitable approximation of the given measure. As an important tool, we obtain a refinement of Anzellotti-Giaquinta approximation for $BV$ functions, which is of separate interest in itself and, in the context of Anzellotti's pairing theory for divergence-measure fields, implies a new way of approximating $λ$-pairings, as well as new bounds for their total variation. These results are also relevant due to their application in the study of weak solutions to the non-parametric prescribed mean curvature equation with measure data, which is explored in a subsequent work.

math.AP↗
arXiv2024

The prescribed mean curvature measure equation in non-parametric form

We introduce a weak formulation of the non-parametric prescribed mean curvature equation with measure data and show the existence and several properties of $BV$ solutions under natural assumptions on the prescribed measure. Our approach does not rely on approximate or viscosity-type solutions. It requires combining various ingredients, including Anzellotti's pairing theory for divergence-measure fields and its recent developments, a refinement of Anzellotti-Giaquinta approximation, and convex duality theory. We also prove a Gamma-convergence result valid for suitable smooth approximations of the prescribed measure, and a maximum principle for continuous weak solutions. We finally construct some examples of non-uniqueness, showing at the same time the need for the continuity assumption in the maximum principle and an unexpected feature of weak solutions.

math.AP↗
arXiv2022

Emergence of Wulff-Crystals from atomistic systems on the FCC and HCP lattices

We consider a system of $N$ hard spheres sitting on the nodes of either the $\mathrm{FCC}$ or $\mathrm{HCP}$ lattice and interacting via a sticky-disk potential. As $N$ tends to infinity (continuum limit), assuming the interaction energy does not exceed that of the ground-state by more than $N^{2/3}$ (surface scaling), we obtain the variational coarse grained model by $Γ$-convergence. More precisely, we prove that the continuum limit energies are of perimeter type and we compute explicitly their Wulff shapes. Our analysis shows that crystallization on $\mathrm{FCC}$ is preferred to that on $\mathrm{HCP}$ for $N$ large enough. The method is based on integral representation and concentration-compactness results that we prove for general periodic lattices in any dimension.

math.AP↗
arXiv2022

Training Quantised Neural Networks with STE Variants: the Additive Noise Annealing Algorithm

Training quantised neural networks (QNNs) is a non-differentiable optimisation problem since weights and features are output by piecewise constant functions. The standard solution is to apply the straight-through estimator (STE), using different functions during the inference and gradient computation steps. Several STE variants have been proposed in the literature aiming to maximise the task accuracy of the trained network. In this paper, we analyse STE variants and study their impact on QNN training. We first observe that most such variants can be modelled as stochastic regularisations of stair functions; although this intuitive interpretation is not new, our rigorous discussion generalises to further variants. Then, we analyse QNNs mixing different regularisations, finding that some suitably synchronised smoothing of each layer map is required to guarantee pointwise compositional convergence to the target discontinuous function. Based on these theoretical insights, we propose additive noise annealing (ANA), a new algorithm to train QNNs encompassing standard STE and its variants as special cases. When testing ANA on the CIFAR-10 image classification benchmark, we find that the major impact on task accuracy is not due to the qualitative shape of the regularisations but to the proper synchronisation of the different STE variants used in a network, in accordance with the theoretical results.

cs.LG↗
arXiv2022

The isoperimetric problem in $2$d domains without necks

We give a complete characterization of all isoperimetric sets contained in a domain of the Euclidean plane, that is bounded by a Jordan curve and satisfies a no-neck property. Further, we prove that the isoperimetric profile of such domain is convex above the volume of the largest ball contained in it, and that its square is globally convex.

math.AP↗
arXiv2020

Analytical aspects of non-differentiable neural networks

Research in computational deep learning has directed considerable efforts towards hardware-oriented optimisations for deep neural networks, via the simplification of the activation functions, or the quantization of both activations and weights. The resulting non-differentiability (or even discontinuity) of the networks poses some challenging problems, especially in connection with the learning process. In this paper, we address several questions regarding both the expressivity of quantized neural networks and approximation techniques for non-differentiable networks. First, we answer in the affirmative the question of whether QNNs have the same expressivity as DNNs in terms of approximation of Lipschitz functions in the $L^{\infty}$ norm. Then, considering a continuous but not necessarily differentiable network, we describe a layer-wise stochastic regularisation technique to produce differentiable approximations, and we show how this approach to regularisation provides elegant quantitative estimates. Finally, we consider networks defined by means of Heaviside-type activation functions, and prove for them a pointwise approximation result by means of smooth networks under suitable assumptions on the regularised activations.

cs.LG↗
arXiv2020

Minimizers of the prescribed curvature functional in a Jordan domain with no necks

We provide a geometric characterization of the minimal and maximal minimizer of the prescribed curvature functional $P(E)-κ|E|$ among subsets of a Jordan domain $Ω$ with no necks of radius $κ^{-1}$, for values of $κ$ greater than or equal to the Cheeger constant of $Ω$. As an application, we describe all minimizers of the isoperimetric profile for volumes greater than the volume of the minimal Cheeger set, relative to a Jordan domain $Ω$ which has no necks of radius $r$, for all $r$. Finally, we show that for such sets and volumes the isoperimetric profile is convex.

math.AP↗
arXiv2020

Rigidity and trace properties of divergence-measure vector fields

We consider a $φ$-rigidity property for divergence-free vector fields in the Euclidean $n$-space, where $φ(t)$ is a non-negative convex function vanishing only at $t=0$. We show that this property is always satisfied in dimension $n=2$, while in higher dimension it requires some further restriction on $φ$. In particular, we exhibit counterexamples to \textit{quadratic rigidity} (i.e., when $φ(t) = ct^2$) in dimension $n\ge 4$. The validity of the quadratic rigidity, which we prove in dimension $n=2$, implies the existence of the trace of a divergence-measure vector field $ξ$ on a $\mathcal{H}^{1}$-rectifiable set $S$, as soon as its weak normal trace $[ξ\cdot ν_S]$ is maximal on $S$. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.

math.AP↗
arXiv2020

Weak and approximate curvatures of a measure: a varifold perspective

By revisiting the notion of generalized second fundamental form originally introduced by Hutchinson for a special class of integral varifolds, we define a weak curvature tensor that is particularly well-suited for being extended to general varifolds of any dimension and codimension through regularization. The resulting approximate second fundamental forms are defined not only for piecewise-smooth surfaces, but also for datasets of very general type (like, e.g., point clouds). We obtain explicitly computable formulas for both weak and approximate curvature tensors, we exhibit structural properties and prove convergence results, and lastly we provide some numerical tests on point clouds that confirm the generality and effectiveness of our approach.

math.CA↗
arXiv2019

Additive Noise Annealing and Approximation Properties of Quantized Neural Networks

We present a theoretical and experimental investigation of the quantization problem for artificial neural networks. We provide a mathematical definition of quantized neural networks and analyze their approximation capabilities, showing in particular that any Lipschitz-continuous map defined on a hypercube can be uniformly approximated by a quantized neural network. We then focus on the regularization effect of additive noise on the arguments of multi-step functions inherent to the quantization of continuous variables. In particular, when the expectation operator is applied to a non-differentiable multi-step random function, and if the underlying probability density is differentiable (in either classical or weak sense), then a differentiable function is retrieved, with explicit bounds on its Lipschitz constant. Based on these results, we propose a novel gradient-based training algorithm for quantized neural networks that generalizes the straight-through estimator, acting on noise applied to the network's parameters. We evaluate our algorithm on the CIFAR-10 and ImageNet image classification benchmarks, showing state-of-the-art performance on AlexNet and MobileNetV2 for ternary networks.

cs.LG↗
arXiv2018

The Cheeger constant of curved tubes

We compute the Cheeger constant of spherical shells and tubular neighbourhoods of complete curves in an arbitrary dimensional Euclidean space.

math.OC↗