arXiv subjects

Matteo Giordano

Publications and source records attributed to Matteo Giordano.

At least 19 recordsLinked to original sources

arXiv2026

Posterior contraction rates in Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families

We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a standard Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on the Fisher information and suitable local regularity assumptions, we show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth. Our analysis builds on the novel approach to posterior contraction based on the Wasserstein distance recently introduced by Dolera et al. (2024). It combines refined Laplace-type estimates for infinite-dimensional integrals associated to the posterior kernels with a mixed-geometry estimate controlling their stability under fluctuations in the data, itself resting on a tailored Poincar\'e inequality for posterior distributions conditioned on neighbourhoods of the truth. We apply the general theory to density estimation with a logistic parametrisation, Poisson intensity estimation with an exponential link, and the Gaussian white-noise model, yielding minimax contraction rates in Sobolev norms across all three settings. In particular, these yield optimal recovery of density score functions and derivatives of Poisson intensities.

math.ST↗

arXiv2026

Increasing domain asymptotics for covariate-based nonparametric Bayesian intensity estimation with Gaussian and Besov-Laplace priors

We study the problem of estimating the intensity function of a covariate-driven point process based on observations of the points and covariates over a large window. We consider the nonparametric Bayesian approach, and show that a wide class of Gaussian priors, combined with flexible link functions, achieves minimax-optimal posterior contraction rates in the increasing domain asymptotics and under the assumption that the covariates be ergodic. We also employ Besov-Laplace priors, which are popular in imaging and inverse problems due to their edge-preserving and sparsity-promoting properties. We prove that these yield optimal estimation of spatially inhomogeneous intensities belonging to Besov spaces with low integrability index. These results are based on a general concentration theorem that extends recent findings from the literature. To corroborate the theory, we provide extensive numerical simulations, implementing the considered procedures via suitable posterior sampling schemes. Further, we present two real data analyses motivated by applications in forestry and the environmental sciences.

math.ST↗

arXiv2026

Dirac mode localization in QCD near the crossover temperature

We study the localization properties of the low-lying Dirac eigenmodes in QCD near the crossover temperature, using stout-smeared staggered fermions and Symanzik-improved gauge action on the lattice. On $N_{\mathrm{t}}=8$ lattices we find that localized low modes, absent at low temperature, appear at a temperature $T_{\mathrm{loc}}$ in the range $150\,\mathrm{MeV}\le T_{\mathrm{loc}}\le 160\,\mathrm{MeV}$, well within the chiral crossover range as determined from the chiral condensate and from the light-quark susceptibility. Since with our choice of lattice action neither the chiral transition region nor the renormalized mobility edges change significantly above $N_{\mathrm{t}}=8$, our conclusion that $T_{\mathrm{loc}}$ is in the chiral crossover region is expected to remain valid in the continuum limit.

hep-lat↗

arXiv2025

Hierarchical Besov-Laplace priors for spatially inhomogeneous binary classification

We study nonparametric Bayesian binary classification, in the case where the unknown probability response function is possibly spatially inhomogeneous, for example, being generally flat across the domain but presenting localized sharp variations. We consider a hierarchical procedure based on the Besov-Laplace priors from the inverse problems and imaging literature, with a carefully tuned hyper-prior on the regularity parameter. We show that the resulting posterior distribution concentrates towards the ground truth at optimal rate, automatically adapting to the unknown regularity. To implement posterior inference in practice, we devise an efficient Markov chain Monte Carlo (MCMC) algorithm based on recent ad-hoc dimension-robust methods for Besov-Laplace priors. We then test the considered approach in extensive numerical simulations, where we obtain a solid corroboration of the theoretical results.

math.ST↗

arXiv2025

Dirac spectrum in the chirally symmetric phase of a gauge theory. I

I study the consequences of chiral symmetry restoration for the Dirac spectrum in finite-temperature gauge theories in the two-flavor chiral limit, using Ginsparg--Wilson fermions on the lattice. I prove that chiral symmetry is restored at the level of the susceptibilities of scalar and pseudoscalar bilinears if and only if all these susceptibilities do not diverge in the chiral limit $m\to 0$, with $m$ the common mass of the light fermions. This implies in turn that they are infinitely differentiable functions of $m^2$ at $m=0$, or $m$ times such a function, depending on whether they contain an even or odd number of isosinglet bilinears. Expressing scalar and pseudoscalar susceptibilities in terms of the Dirac spectrum, I use their finiteness in the symmetric phase to derive constraints on the spectrum, and discuss the resulting implications for the fate of the anomalous $\mathrm{U}(1)_A$ symmetry in the chiral limit. I also discuss the differentiability properties of spectral quantities with respect to $m^2$, and show from first principles that the topological properties of the theory in the chiral limit are characterized by an instanton gas-like behavior if $\mathrm{U}(1)_A$ remains effectively broken.

hep-lat↗

arXiv2025

Dirac spectrum in the chirally symmetric phase of a gauge theory. II

I discuss the consequences of the constraints imposed on the Dirac spectrum by the restoration of chiral symmetry in the chiral limit of gauge theories with two light fermion flavors, with particular attention to the fate of the anomalous $\mathrm{U}(1)_A$ symmetry. Under general, physically motivated assumptions on the spectral density and on the two-point eigenvalue correlation function, I show that effective $\mathrm{U}(1)_A$ breaking in the symmetric phase requires specific spectral features, including a spectral density effectively behaving as $m^2\delta(\lambda)$ in the chiral limit, a two-point function singular at zero, and delocalized near-zero modes, besides an instanton gas-like behavior of the topological charge distribution. I then discuss a $\mathrm{U}(1)_A$-breaking scenario characterized by a power-law divergent spectral peak tending to $O(m^4)/|\lambda|$ in the chiral limit and by a near-zero mobility edge, and argue that the mixing of the approximate zero modes associated with a dilute gas of topological objects provides a concrete physical mechanism producing the required spectral features, and so a viable mechanism for effective $\mathrm{U}(1)_A$ breaking in the symmetric phase of a gauge theory.

hep-lat↗

arXiv2025

Comment on "Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature"

Aoki, Fukaya, and Taniguchi claim that both the spectral density of the Dirac operator at the origin and the topological susceptibility must vanish identically for sufficiently small but nonzero quark mass $m$ in the chirally symmetric phase of quantum chromodynamics with two light quark flavors, under certain technical assumptions on the spectrum and on the dependence of observables on $m$. I argue that a crucial step of their proof is not justified, and the validity of these conclusions should be reassessed. In an additional note I refute the objections raised by S. Aoki and H. Fukaya in their reply to my comment. I show that both the $m^2$-analyticity assumption they invoke against my arguments, and the very claim that these arguments criticize, imply that all local gluonic correlators are independent of the light-quark mass at small mass. This behavior, however, is not observed in quantum chromodynamics. I then show that their claim of having found a mistake in my arguments is baseless.

hep-lat↗

arXiv2025

Bayesian inference with Besov-Laplace priors for spatially inhomogeneous binary classification surfaces

In this article, we study the binary classification problem with supervised data, in the case where the covariate-to-probability-of-success map is possibly spatially inhomogeneous. We devise nonparametric Bayesian procedures with Besov-Laplace priors, which are prior distributions on function spaces routinely used in imaging and inverse problems in view of their useful edge-preserving and sparsity-promoting properties. Building on a recent line of work in the literature, we investigate the theoretical asymptotic recovery properties of the associated posterior distributions, and show that suitably tuned Besov-Laplace priors lead to minimax-optimal posterior contraction rates as the sample size increases, under the frequentist assumption that the data have been generated by a spatially inhomogeneous ground truth belonging to a Besov space.

math.ST↗

arXiv2025

A nonparametric Bayesian analysis of independent and identically distributed observations of covariate-driven Poisson processes

An important task in the statistical analysis of inhomogeneous point processes is to investigate the influence of a set of covariates on the point-generating mechanism. In this article, we consider the nonparametric Bayesian approach to this problem, assuming that $n$ independent and identically distributed realizations of the point pattern and the covariate random field are available. In many applications, different covariates are often vastly diverse in physical nature, resulting in anisotropic intensity functions whose variations along distinct directions occur at different smoothness levels. To model this scenario, we employ hierarchical prior distributions based on multi-bandwidth Gaussian processes. We prove that the resulting posterior distributions concentrate around the ground truth at optimal rate as $n\to\infty$, and achieve automatic adaptation to the anisotropic smoothness. Posterior inference is concretely implemented via a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm that incorporates a dimension-robust sampling scheme to handle the functional component of the proposed nonparametric Bayesian model. Our theoretical results are supported by extensive numerical simulation studies. Further, we present an application to the analysis of a Canadian wildfire dataset.

stat.ME↗

arXiv2025

A comparative analysis of machine learning algorithms for predicting probabilities of default

Predicting the probability of default (PD) of prospective loans is a critical objective for financial institutions. In recent years, machine learning (ML) algorithms have achieved remarkable success across a wide variety of prediction tasks; yet, they remain relatively underutilised in credit risk analysis. This paper highlights the opportunities that ML algorithms offer to this field by comparing the performance of five predictive models-Random Forests, Decision Trees, XGBoost, Gradient Boosting and AdaBoost-to the predominantly used logistic regression, over a benchmark dataset from Scheule et al. (Credit Risk Analytics: The R Companion). Our findings underscore the strengths and weaknesses of each method, providing valuable insights into the most effective ML algorithms for PD prediction in the context of loan portfolios.

q-fin.RM↗

arXiv2025

Bayesian Inference for Initial Heat States with Gaussian Series Priors

We consider the statistical linear inverse problem of recovering the unknown initial heat state from noisy interior measurements over an inhomogeneous domain of the solution to the heat equation at a fixed time instant. We employ nonparametric Bayesian procedures with Gaussian series priors defined on the Dirichlet-Laplacian eigenbasis, yielding convenient conjugate posterior distributions with explicit expressions for posterior inference. We review recent theoretical results that provide asymptotic performance guarantees (in the large sample size limit) for the resulting posterior-based point estimation and uncertainty quantification. We further provide an implementation of the approach, and illustrate it via a numerical simulation study.

stat.ME↗

arXiv2025

Gaussian Process Methods for Covariate-Based Intensity Estimation

We study nonparametric Bayesian inference for the intensity function of a covariate-driven point process. We extend recent results from the literature, showing that a wide class of Gaussian priors, combined with flexible link functions, achieve minimax optimal posterior contraction rates. Our result includes widespread prior choices such as the popular Mat\'ern processes, with the standard exponential (and sigmoid) link, and implies that the resulting methodologically attractive procedures optimally solve the statistical problem at hand, in the increasing domain asymptotics and under the common assumption in spatial statistics that the covariates are stationary and ergodic.

math.ST↗

arXiv2025

Semiparametric Bernstein-von Mises theorems for reversible diffusions

We establish a general semiparametric Bernstein-von Mises theorem for Bayesian nonparametric priors based on continuous observations in a periodic reversible multidimensional diffusion model. We consider a wide range of functionals satisfying an approximate linearization condition, including several nonlinear functionals of the invariant measure. Our result is applied to Gaussian and Besov-Laplace priors, showing these can perform efficient semiparametric inference and thus justifying the corresponding Bayesian approach to uncertainty quantification. Our theoretical results are illustrated via numerical simulations.

math.ST↗

arXiv2025

Localization in the Dirac spectrum and gauge-field topology

We study localization of the low Dirac modes in 3+1 dimensional pure $\mathrm{SU}(3)$ gauge theory at zero and nonzero imaginary $\theta$ angle, with the aim of better characterizing the relation between low-mode localization and topological features of gauge theories. We show that the mobility edge observed in the deconfined phase at $\theta=0$ is present also at nonzero $\theta$, appearing exactly at the deconfinement transition. We find that the mobility edge is affected by topology only indirectly, through its effects on the ordering of the Polyakov loop. Moreover, the change in the mobility edge is strongly correlated with the change in the Polyakov-loop expectation value, both as one moves along the critical line, and as one departs from it toward higher temperatures. This further strengthens the connection between low-mode localization and deconfinement, showing in particular the key role played by the ordering of the Polyakov loop.

hep-lat↗

arXiv2025

Localization of Dirac modes in a finite temperature SU(2) Higgs model

Low-lying Dirac modes become localized at the finite-temperature transition in QCD and other gauge theories, indicating a strong connection between localization and deconfinement. This phenomenon can be understood through the "sea/islands" picture: in the deconfined phase, modes become trapped on "islands" of Polyakov loop fluctuations within a "sea" of ordered Polyakov loops. To test the universality of the "sea/islands" mechanism, we investigate whether changes in the localization properties of low modes occur across other thermal transitions where the Polyakov loop becomes ordered, beyond the usual deconfinement transition. The fixed-length SU(2)-Higgs model is appropriate for this study. After mapping out the phase diagram, we find that low Dirac modes become localized in the deconfined and Higgs phases, where the Polyakov loop is ordered. However, localization is absent in the confined phase. These findings confirm the "sea/islands" picture of localization.

hep-lat↗

arXiv2024

Constraints on the Dirac spectrum from chiral symmetry restoration and the fate of $\mathrm{U}(1)_A$ symmetry

I discuss chiral symmetry restoration in the chiral limit $m\to 0$ of QCD with two light quark flavours of mass $m$, focussing on its consequences for scalar and pseudoscalar susceptibilities, and on the resulting constraints on the Dirac spectrum. I show that $\mathrm{U}(1)_A$ symmetry remains broken in the $\mathrm{SU}(2)_A$ symmetric phase if the spectral density $\rho(\lambda;m)$ develops a singular near-zero peak, tending to $O(m^4)/\lambda$ in the chiral limit. Moreover, $\mathrm{SU}(2)_A$ restoration requires that the number of modes in the peak be proportional to the topological susceptibility, indicating that such a peak must be of topological origin.

hep-lat↗

arXiv2024

On strong posterior contraction rates for Besov-Laplace priors in the white noise model

In this article, we investigate the problem of estimating a spatially inhomogeneous function and its derivatives in the white noise model using Besov-Laplace priors. We show that smoothness-matching priors attains minimax optimal posterior contraction rates, in strong Sobolev metrics, over the Besov spaces $B^\beta_{11}$, $\beta > d/2$, closing a gap in the existing literature. Our strong posterior contraction rates also imply that the posterior distributions arising from Besov-Laplace priors with matching regularity enjoy a desirable plug-in property for derivative estimation, entailing that the push-forward measures under differential operators optimally recover the derivatives of the unknown regression function. The proof of our results relies on the novel approach to posterior contraction rates, based on Wasserstein distance, recently developed by Dolera, Favaro and Mainini (Probability Theory and Related Fields, 2024). We show how this approach allows to overcome some technical challenges that emerge in the frequentist analysis of smoothness-matching Besov-Laplace priors.

math.ST↗

arXiv2024

Center vortices and localized Dirac modes in the deconfined phase of (2+1)-dimensional lattice $\mathbb{Z}_2$ gauge theory

We study the deconfinement transition in (2+1)-dimensional lattice $\mathbb{Z}_2$ gauge theory both as a percolation transition of center vortices and as a localization transition for the low-lying Dirac modes. We study in detail the critical properties of the Anderson transition in the Dirac spectrum in the deconfined phase, showing that it is of BKT type; and the critical properties of the center-vortex percolation transition, showing that they differ from those of ordinary two-dimensional percolation. We then study the relation between localized modes and center vortices in the deconfined phase, identifying the simple center-vortex structures that mainly support the localized Dirac modes. As the system transitions to the confined phase, center vortices merge together into an infinite cluster, causing the low Dirac modes to delocalize.

hep-lat↗