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Giacomo Graziani

Publications and source records attributed to Giacomo Graziani.

14 recordsLinked to original sources
arXiv2026

On the Euclidean Distance Degree of Shallow Polynomial Networks

We study the projective geometry of shallow polynomial neural networks, namely families of homogeneous polynomial maps realised by a single hidden layer with monomial activation. The closure of the set of maps realised by such a network is a projective variety, its neurovariety. A natural measure of the algebraic complexity of the corresponding best-approximation problem is the generic Euclidean distance degree $\mathrm{gED}$, which counts the complex critical points of the squared distance from a general point. For fixed input dimension, width and activation degree, we prove that $\mathrm{gED}$ is eventually polynomial in the output dimension. Its degree and leading coefficient are governed by an auxiliary variety parametrising the spaces spanned by powers of linear forms. When the activation degree is at least the width, this extends: on the same stable range, $\mathrm{gED}$ is a polynomial in the output dimension and the activation degree jointly. Our proof constructs a proper birational model of the conormal variety over a base that does not depend on the output dimension, and expresses $\mathrm{gED}$ through Chern classes on it. We give closed formulas in two basic cases and determine the leading term for width two.

math.AG↗
arXiv2026

Algebraic Networks and Architectural Degenerations

We study the geometry of polynomial neural networks with monomial activation functions and no bias. We introduce a general framework of algebraic networks, together with their realization maps and associated affine neurovarieties. In this setting we define morphisms, subnetworks, symmetry groups and quotient parameter spaces and we discuss geometric notions of identifiability and reducibility. Our main goal is to relate the singularities of neurovarieties to degenerations of the underlying architecture. For fully connected networks, we define the architectural degeneracy locus as the locus of functions admitting a representation by parameters with a rank-deficient layer or an inactive hidden neuron. We prove that, for fully connected networks with non-increasing widths and scalar output, full parameters give smooth points of the corresponding neurovariety under explicit layerwise regularity assumptions. In particular, for these architectures, the singular locus is contained in the architectural degeneracy locus.

math.AG↗
arXiv2025

On the Euclidean Distance Degree of Quadratic Two-Neuron Neural Networks

We study the Euclidean Distance degree of algebraic neural network models from the perspective of algebraic geometry. Focusing on shallow networks with two neurons, quadratic activation, and scalar output, we identify the associated neurovariety with the second secant variety of a quadratic Veronese embedding. We introduce and analyze the virtual Euclidean Distance degree, a projective invariant defined as the sum of the polar degrees of the variety, which coincides with the usual Euclidean Distance degree for a generic choice of scalar product. Using intersection theory, Chern-Mather classes, and the Nash blow-up provided by Kempf's resolution, we reduce the computation of the virtual Euclidean Distance degree to explicit intersection numbers on a Grassmannian. Applying equivariant localization, we prove that this invariant depends stably polynomially on the input dimension. Numerical experiments based on homotopy continuation illustrate the dependence of the Euclidean Distance degree on the chosen metric and highlight the distinction between the generic and nongeneric cases, such as the Bombieri-Weyl metric.

math.AG↗
arXiv2025

Grothendieck's proof of Hirzebruch-Riemann-Roch theorem

The Riemann-Roch Theorem is one of the cornerstones of algebraic geometry, connecting algebraic data (sheaf cohomology) with geometric ones (intersection theory). This survey paper provides a self-contained introduction and a complete proof of the Hirzebruch-Riemann-Roch (HRR) Theorem for smooth projective varieties over an algebraically closed field. Starting from the classical formulations for curves and surfaces, we introduce the two modern tools necessary for the generalization: the Grothendieck group $K_{0}(X)$ as the natural setting for the Euler characteristic, and the Chow ring $A_{\bullet}(X)$ as the setting for cycles and intersection theory. We then construct the fundamental bridge between these two worlds\textemdash the Chern character ($\mathrm{ch}$) and the Todd class ($\mathrm{td}$) \textemdash culminating in a full proof of the HRR formula: \[ χ(X,\mathcal{E})=\int_{X}\mathrm{ch}(\mathcal{E})\cdot\mathrm{td}(X) \] We conclude by showing how this general formula recovers the classical theorems for curves and surfaces.

math.AG↗
arXiv2025

Ruled and rational surfaces and their models

One of the most powerful ideas in the study and classification of algebraic varieties is the notion of a model: that is, to single out an object, in the appropriate isomorphism class, with nice properties. This survey aims to define and study suitable models of rational and, more generally, ruled surfaces in the smooth complex case, and to use them to study such surfaces.

math.AG↗
arXiv2025

Road map for the tuning of hadronic interaction models with accelerator-based and astroparticle data

In high-energy and astroparticle physics, event generators play an essential role, even in the simplest data analyses. As analysis techniques become more sophisticated, e.g. based on deep neural networks, their correct description of the observed event characteristics becomes even more important. Physical processes occurring in hadronic collisions are simulated within a Monte Carlo framework. A major challenge is the modeling of hadron dynamics at low momentum transfer, which includes the initial and final phases of every hadronic collision. QCD-inspired phenomenological models used for these phases cannot guarantee completeness or correctness over the full phase space. These models usually include parameters which must be tuned to suitable experimental data. Until now, event generators have been developed and tuned mainly on the basis of data from high-energy physics experiments at accelerators. The wealth of data available from the latest generation of astroparticle experiments has not yet been fully exploited, and in many cases is not satisfactorily described. Both kinds of data sets are complementary as astroparticle experiments provide sensitivity especially to hadrons produced nearly parallel to the collision axis and cover center-of-mass energies up to several hundred TeV, well beyond those reached at colliders so far. In this report, we provide an overview of state-of-the-art event generators and their tuning, including the most relevant inputs from high-energy accelerator and astroparticle experiments. We present a road map that shows, for the first time, how the unified tuning of event generators with accelerator-based and astroparticle data can be performed.

astro-ph.HE↗
arXiv2022

A Neural-Network-defined Gaussian Mixture Model for particle identification applied to the LHCb fixed-target programme

Particle identification in large high-energy physics experiments typically relies on classifiers obtained by combining many experimental observables. Predicting the probability density function (pdf) of such classifiers in the multivariate space covering the relevant experimental features is usually challenging. The detailed simulation of the detector response from first principles cannot provide the reliability needed for the most precise physics measurements. Data-driven modelling is usually preferred, though sometimes limited by the available data size and different coverage of the feature space by the control channels. In this paper, we discuss a novel approach to the modelling of particle identification classifiers using machine-learning techniques. The marginal pdf of the classifiers is described with a Gaussian Mixture Model, whose parameters are predicted by Multi Layer Perceptrons trained on calibration data. As a proof of principle, the method is applied to the data acquired by the LHCb experiment in its fixed-target configuration. The model is trained on a data sample of proton-neon collisions and applied to smaller data samples of proton-helium and proton-argon collisions collected at different centre-of-mass energies. The method is shown to perform better than a detailed simulation-based approach, to be fast and suitable to be applied to a large variety of use cases.

hep-ex↗
arXiv2020

Muon identification for LHCb Run 3

Muon identification is of paramount importance for the physics programme of LHCb. In the upgrade phase, starting from Run 3 of the LHC, the trigger of the experiment will be solely based on software. The luminosity increase to $2\times10^{33}$ cm$^{-2}$s$^{-1}$ will require an improvement of the muon identification criteria, aiming at performances equal or better than those of Run 2, but in a much more challenging environment. In this paper, two new muon identification algorithms developed in view of the LHCb upgrade are presented, and their performance in terms of signal efficiency versus background reduction is shown.

hep-ex↗
arXiv2019

Results on heavy ion physics at LHCb

In the last years, the \lhcb experiment established itself as an important contributor to heavy ion physics by exploiting some of its specific features. Production of particles, notably heavy flavour states, can be studied in p-p, p-Pb and Pb-Pb collisions at LHC energies in the forward rapidity region (pseudorapidity between 2 and 5), providing measurements which are highly complementary to the other LHC experiments. Moreover, owing to its forward geometry, the detector is also well suited to study fixed-target collisions, obtained by impinging the LHC beams on gas targets with different mass numbers. In this configuration, p-A collisions can be studied at the relatively unexplored scale of sqrt(sNN) ~ 100 GeV, also providing valuable inputs to cosmic ray physics. An overview of the measurements obtained so far by the LHCb ion program is presented.

hep-ex↗
arXiv2017

Fixed target measurements at LHCb for cosmic rays physics

The LHCb experiment has the unique possibility, among the LHC experiments, to be operated in fixed target mode, using its internal gas target. The energy scale achievable at the LHC, combined with the LHCb forward geometry and detector capabilities, allow to explore particle production in a wide Bjorken-$x$ range at the $\sqrt{s_{\scriptscriptstyle\rm NN}} \sim 100$ GeV energy scale, providing novel inputs to nuclear and cosmic ray physics. The first measurement of antiproton production in collisions of LHC protons on helium nuclei at rest is presented. The knowledge of this cross-section is of great importance for the study of the cosmic antiproton flux, and the LHCb results are expected to improve the interpretation of the recent high-precision measurements of cosmic antiprotons performed by the space-borne PAMELA and AMS-02 experiments.

hep-ex↗
arXiv2015

BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras

A BiHom-associative algebra is a (nonassociative) algebra $A$ endowed with two commuting multiplicative linear maps $α,β\colon A\rightarrow A$ such that $α(a)(bc)=(ab)β(c)$, for all $a, b, c\in A$. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).

math.RA↗
arXiv2011

Recent LHCb Results

The LHCb experiment started its physics program with the 37/pb of pp collisions at 7 TeV c.m. energy delivered by the LHC during 2010. The performances and capability of the experiment, conceived for precision measurements in the heavy flavour sector, are illustrated through the first results from the experimental core program. A rich set of production studies provide precision QCD and EW tests in the unique high rapidity region covered by LHCb. Notably, results for W and Z production are very encouraging for setting constraints on the parton PDFs.

hep-ex↗
arXiv2002

Latest Results on Kaon Physics from the NA48 Experiment

The NA48 experiment, conceived primarily to look for direct CP violation in neutral kaon decays, has recently published the so far most precise determination of the epsilon'/epsilon parameter. After reviewing shortly this result, we report on the 2001 data-taking, which concluded the epsilon' program by collecting a substantial amount of data with different beam intensity conditions. We also present new precision measurements of the K0 and eta masses and of the K-short lifetime, that provide consistency checks of our analysis. Finally, the prospects for the future experimental program are discussed.

hep-ex↗