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Claudio Durastanti

Publications and source records attributed to Claudio Durastanti.

At least 19 recordsLinked to original sources
arXiv2026

Exploring the Structure of Anisotropic Random Fields on the Sphere

This paper addresses fundamental questions about the existence and structure of isotropic and anisotropic spherical random fields and the properties of composite transformations of isotropic fields. It introduces a hierarchical structure comprising several classes of anisotropic random fields, using geometric and spectral approaches, and provides a geometric--spectral classification of second-order anisotropy on $\mathbb{S}^2$. We characterize feasible isotropic and anisotropic scenarios and identify those that, in principle, cannot occur, in contrast to the case of random fields in Euclidean space. Furthermore, we establish connections between geometric invariance properties and the structure of the spectral covariance matrix of spherical random fields. The obtained results lay the foundations for applications in statistics and data analysis, providing guidance on feasible models, their spectral representations, and appropriate sampling schemes for parameter estimation or anisotropy hypothesis testing for spherical data.

math.PR↗
arXiv2026

Adaptive thresholding for wavelet-based nonparametric heteroskedastic variance estimation on the sphere

This paper addresses the nonparametric estimation of a spatially varying, heteroskedastic variance function on the unit sphere within a regression framework. While adaptive regression estimation is well-established on manifolds, characterizing localized noise structures presents unique theoretical obstacles due to bias propagation from the unknown mean function. To circumvent this, we propose a fully data-driven, multiresolution estimator based on localized spherical frames, namely, needlets, combined with a hard-thresholding protocol and a sample-splitting scheme. The approach exploits the excellent spatial and frequency localization properties of needlets to adaptively capture the local features of the variance function. We prove that the proposed estimator achieves the minimax-optimal rate of convergence over spherical Besov spaces under standard loss functions, exhibiting spatial adaptivity without requiring prior knowledge of the regularity of the variance function. This adaptivity highlights the efficacy of the method in analyzing spherical data characterized by complex heteroskedastic errors, with potential applications in fields such as cosmology, environmental modeling, and geophysics

math.ST↗
arXiv2026

Gaussian approximation for non-linearity parameter estimation in perturbed random fields on the sphere

We develop a probabilistic framework for the asymptotic analysis of a bispectrum-based estimator of primordial non-Gaussianity for isotropic random fields on the sphere in the high-resolution regime. By reformulating the estimation problem as an ordinary least squares regression, we derive the asymptotic moments of the estimator. Combining these results with Stein-Malliavin techniques on Wiener chaos yields a quantitative Gaussian approximation with an explicit convergence rate in total variation distance. The analysis relies on sharp asymptotic estimates for the deterministic weights arising from spherical harmonic coupling coefficients. Numerical experiments illustrate the predicted scaling laws and provide qualitative evidence for the asymptotic Gaussian behavior.

math.ST↗
arXiv2026

Spectral Diffusion Models on the Sphere

Diffusion models provide a principled framework for generative modeling via stochastic differential equations and time-reversed dynamics. However, extension of spectral diffusion approaches to spherical data raises nontrivial geometric and stochastic issues that are absent in the Euclidean setting. In this work, we develop a diffusion modeling framework defined directly on finite-dimensional spherical harmonic representations of real-valued functions on the sphere. We show that the spherical discrete Fourier transform maps spatial Brownian motion to a constrained Gaussian process in the frequency domain with deterministic, generally non-isotropic covariance. This induces modified forward- and reverse-time stochastic differential equations in the spectral domain. As a consequence, spatial and spectral score matching objectives are generally no longer equivalent, even in the band-limited setting. We establish a quantitative relationship between the two objectives, showing that the geometry-induced covariance of the spectral noise gives rise to a distinct, geometry-dependent inductive bias. We also derive the corresponding forward and reverse diffusion equations and characterize the induced noise covariance.

math.PR↗
arXiv2026

Adaptive estimation of Sobolev-type energy functionals on the sphere

We study the estimation of quadratic Sobolev-type integral functionals of an unknown density on the unit sphere. The functional is defined through fractional powers of the Laplace--Beltrami operator and provides a global measure of smoothness and spectral energy. Our approach relies on spherical needlet frames, which yield a localized multiscale decomposition while preserving tight frame properties in the natural square-integrable function space on the sphere. We construct unbiased estimators of suitably truncated versions of the functional and derive sharp oracle risk bounds through an explicit bias--variance analysis. When the smoothness of the density is unknown, we propose a Lepski-type data-driven selection of the resolution level. The resulting adaptive estimator achieves minimax-optimal rates over Sobolev classes, without resorting to nonlinear or sparsity-based methods.

math.ST↗
arXiv2026

Spectral Bayesian Regression on the Sphere

We develop a fully intrinsic Bayesian framework for nonparametric regression on the unit sphere based on isotropic Gaussian field priors and the harmonic structure induced by the Laplace-Beltrami operator. Under uniform random design, the regression model admits an exact diagonalization in the spherical harmonic basis, yielding a Gaussian sequence representation with frequency-dependent multiplicities. Exploiting this structure, we derive closed-form posterior distributions, optimal spectral truncation schemes, and sharp posterior contraction rates under integrated squared loss. For Gaussian priors with polynomially decaying angular power spectra, including spherical Matérn priors, we establish posterior contraction rates over Sobolev classes, which are minimax-optimal under correct prior calibration. We further show that the posterior mean admits an exact variational characterization as a geometrically intrinsic penalized least-squares estimator, equivalent to a Laplace-Beltrami smoothing spline.

math.ST↗
arXiv2026

Aliasing Effects for Samples of Spin Random Fields on the Sphere

This paper investigates aliasing effects emerging from the reconstruction from discrete samples of spin spherical random fields defined on the two-dimensional sphere. We determine the location in the frequency domain and the intensity of the aliases of the harmonic coefficients in the Fourier decomposition of the spin random field and evaluate the consequences of aliasing errors in the angular power spectrum when the samples of the random field are obtained by using some very popular sampling procedures on the sphere, the equiangular and the Gauss-Jacobi sampling schemes. Finally, we demonstrate that band-limited spin random fields are free from aliases, provided that a sufficiently large number of nodes is used in the selected quadrature rule.

math.PR↗
arXiv2025

Scale Dilation Dynamics in Flexible Bandwidth Needlet Constructions

Flexible bandwidth needlets offer a versatile multiscale framework for analyzing functions on the sphere. A key element in their construction is the dilation sequence, which controls how the multipole consecutive scales are spaced and overlapped. At any resolution level, this sequence determines the center positions of the needlet weight functions and influences their localization in the spatial domain and spectral concentration properties by means of the relative bandwidth ratio. In this paper, we explore the different asymptotic regimes that arise when the dilation sequence exhibits shrinking, stable (standard), or spreading behavior. Moreover, we assume the dilation sequence grows regularly enough to ensure well-defined asymptotic properties. For each regime, we characterize the impact on the geometry of the center scales and the shape of the multipole windows, with particular attention to their overlap structure and spectral coverage. These insights help to clarify the trade-offs between localization, redundancy, and scalability in the design of needlet-type systems, particularly in relation to the study of the asymptotic uncorrelation of needlet coefficients when applied to random fields.

math.ST↗
arXiv2025

Spherical Poisson Needlets with Shrinking Bandwidth

Flexible bandwidth needlets provide a localized multiscale framework with scale-adaptive frequency resolution, enabling effective analysis of spherical Poisson random fields exhibiting spatial inhomogeneity and scale variation. We establish here quantitative Central Limit Theorems for finite-dimensional distributions of spherical Poisson needlets and for the related Poisson needlet coefficients constructed via needlets with shrinking bandwidth on the sphere, and using Stein-Malliavin techniques, we derive explicit rates of normal approximation. In addition, we study the functional convergence of the associated needlet-based random fields. Indeed, our framework provides quantitative control on the limiting behavior in appropriate function spaces. Together, these results offer rigorous probabilistic guarantees for high-resolution spherical data modeling under Poisson sampling.

math.PR↗
arXiv2023

Nonparametric needlet estimation for partial derivatives of a probability density function on the $d$-torus

This paper is concerned with the estimation of the partial derivatives of a probability density function of directional data on the $d$-dimensional torus within the local thresholding framework. The estimators here introduced are built by means of the toroidal needlets, a class of wavelets characterized by excellent concentration properties in both the real and the harmonic domains. In particular, we discuss the convergence rates of the $L^p$-risks for these estimators, investigating on their minimax properties and proving their optimality over a scale of Besov spaces, here taken as nonparametric regularity function spaces.

math.ST↗
arXiv2023

Spherical Poisson Waves

We introduce a model of Poisson random waves in $\mathbb{S}^{2}$ and we study Quantitative Central Limit Theorems when both the rate of the Poisson process and the energy (i.e., frequency) of the waves (eigenfunctions) diverge to infinity. We consider finite-dimensional distributions, harmonic coefficients and convergence in law in functional spaces, and we investigate carefully the interplay between the rates of divergence of eigenvalues and Poisson governing measures.

math.PR↗
arXiv2023

Learning models for classifying Raman spectra of genomic DNA from tumor subtypes

An early detection of different tumor subtypes is crucial for an effective guidance to personalized therapy. While much efforts focus on decoding the sequence of DNA basis to detect the genetic mutations related to cancer, it is becoming clear that physical properties, including structural conformation, stiffness, and shape, as well as biological processes, such as methylation, can be pivotal to recognize DNA modifications. Here we exploit the Surface Enhanced Raman Scattering (SERS) platform, based on disordered silver coated--silicon nanowires, to investigate genomic DNA from subtypes of melanoma and colon cancers and to efficiently discriminate tumor and healthy cells, as well as the different tumor subtypes. The diagnostic information is obtained by performing label--free Raman maps of the dried drops of DNA solutions onto the Ag/NWs mat, and leveraging the classification ability of learning models to reveal the specific and distinct interaction of healthy and tumor DNA molecules with nanowires.

stat.AP↗
arXiv2022

Statistical classification for Raman spectra of tumoral genomic DNA

We exploit Surface-Enhanced Raman Scattering (SERS) to investigate aqueous droplets of genomic DNA deposited onto silver-coated silicon nanowires and we show that it is possible to efficiently discriminate between spectra of tumoral and healthy cells. To assess the robustness of the proposed technique, we develop two different statistical approaches, one based on the Principal Component Analysis of spectral data and one based on the computation of the $\ell^2$ distance between spectra. Both methods prove to be highly efficient and we test their accuracy via the so-called Cohen's $κ$ statistics. We show that the synergistic combination of the SERS spectroscopy and the statistical analysis methods leads to efficient and fast cancer diagnostic applications allowing a rapid and unexpansive discrimination between healthy and tumoral genomic DNA alternative to the more complex and expensive DNA sequencing.

q-bio.QM↗
arXiv2021

Flexible-bandwidth Needlets

We investigate here a generalized construction of spherical wavelets/needlets which admits extra-flexibility in the harmonic domain, i.e., it allows the corresponding support in multipole (frequency) space to vary in more general forms than in the standard constructions. We study the analytic properties of this system and we investigate its behaviour when applied to isotropic random fields: more precisely, we establish asymptotic localization and uncorrelation properties (in the high-frequency sense) under broader assumptions than typically considered in the literature.

math.PR↗
arXiv2020

LASSO estimation for spherical autoregressive processes

The purpose of the present paper is to investigate on a class of spherical functional autoregressive processes in order to introduce and study LASSO (Least Absolute Shrinkage and Selection Operator) type estimators for the corresponding autoregressive kernels, defined in the harmonic domain by means of their spectral decompositions. Some crucial properties for these estimators are proved, in particular, consistency and oracle inequalities.

math.ST↗
arXiv2019

Aliasing effects for random fields over spheres of arbitrary dimension

In this paper, aliasing effects are investigated for random fields defined on the d-dimensional sphere and reconstructed from discrete samples. First, we introduce the concept of an aliasing function on the sphere. The aliasing function allows one to identify explicitly the aliases of a given harmonic coefficient in the Fourier decomposition. Then, we exploit this tool to establish the aliases of the harmonic coefficients approximated by means of the quadrature procedure named spherical uniform sampling. Subsequently, we study the consequences of the aliasing errors in the approximation of the angular power spectrum of an isotropic random field, the harmonic decomposition of its covariance function. Finally, we show that band-limited random fields are aliases-free, under the assumption of a sufficiently large amount of nodes in the quadrature rule.

math.ST↗
arXiv2016

Tail Behaviour of Mexican Needlets

In this paper we study the tail behaviour of Mexican needlets, a class of spherical wavelets introduced by Geller and Mayeli (2009). More specifically, we provide an explicit upper bound depending on the resolution level $j$ and a parameter $s$ governing the shape of the Mexican needlets

math.FA↗