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Domenico Marinucci

Publications and source records attributed to Domenico Marinucci.

At least 55 records · Page 3Linked to original sources
arXiv2016

Needlet estimation of cross-correlation between CMB lensing maps and LSS

In this paper we develop a novel needlet-based estimator to investigate the cross-correlation between cosmic microwave background (CMB) lensing maps and large-scale structure (LSS) data. We compare this estimator with its harmonic counterpart and, in particular, we analyze the bias effects of different forms of masking. In order to address this bias, we also implement a MASTER-like technique in the needlet case. The resulting estimator turns out to have an extremely good signal-to-noise performance. Our analysis aims at expanding and optimizing the operating domains in CMB-LSS cross-correlation studies, similarly to CMB needlet data analysis. It is motivated especially by next generation experiments (such as Euclid) which will allow us to derive much tighter constraints on cosmological and astrophysical parameters through cross-correlation measurements between CMB and LSS.

astro-ph.CO↗
arXiv2016

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

"Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013)). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener-Itô chaos expansion for the nodal length shows that it is dominated by its $4$th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.

math-ph↗
arXiv2016

A Quantitative Central Limit Theorem for the Euler-Poincaré Characteristic of Random Spherical Eigenfunctions

We establish here a Quantitative Central Limit Theorem (in Wasserstein distance) for the Euler-Poincaré Characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler-Poincaré Characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level $u$ is fully degenerate, i.e., the Euler-Poincaré Characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. Our results can be written as an asymptotic second-order Gaussian Kinematic Formula for the excursion sets of Gaussian spherical harmonics.

math.PR↗
arXiv2016

Strong Local Nondeterminism and Exact Modulus of Continuity for Spherical Gaussian Fields

In this paper, we are concerned with sample path properties of isotropic spherical Gaussian fields on $§^2$. In particular, we establish the property of strong local nondeterminism of an isotropic spherical Gaussian field based on the high-frequency behaviour of its angular power spectrum; we then exploit this result to establish an exact uniform modulus of continuity for its sample paths. We also discuss the range of values of the spectral index for which the sample functions exhibit fractal or smooth behaviour.

math.PR↗
arXiv2016

Multiple testing of local maxima for detection of peaks on the (celestial) sphere

We present a topological multiple testing scheme for detecting peaks on the sphere under isotropic Gaussian noise, where tests are performed at local maxima of the observed field filtered by the spherical needlet transform. Our setting is different from the standard Euclidean/large same asymptotic framework, yet highly relevant to realistic experimental circumstances for some important areas of application in astronomy. More precisely, we focus on cases where a single realization of a smooth isotropic Gaussian random field on the sphere is observed, and a number of well-localized signals are superimposed on such background field. The proposed algorithms, combined with the Benjamini-Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency as the signal strength and the frequency of the needlet transform get large. This novel multiple testing method is illustrated in a simulation of point-source detection in Cosmic Microwave Background radiation (CMB) data.

math.ST↗
arXiv2016

High-frequency asymptotics for Lipschitz-Killing curvatures of excursion sets on the sphere

In this paper, we shall be concerned with geometric functionals and excursion probabilities for some nonlinear transforms evaluated on Fourier components of spherical random fields. In particular, we consider both random spherical harmonics and their smoothed averages, which can be viewed as random wavelet coefficients in the continuous case. For such fields, we consider smoothed polynomial transforms; we focus on the geometry of their excursion sets, and we study their asymptotic behaviour, in the high-frequency sense. We focus on the analysis of Euler-Poincaré characteristics, which can be exploited to derive extremely accurate estimates for excursion probabilities. The present analysis is motivated by the investigation of asymmetries and anisotropies in cosmological data. The statistics we focus on are also suitable to deal with spherical random fields which can only be partially observed, the canonical example being provided by the masking effect of the Milky Way on Cosmic Microwave Background (CMB) radiation data.

math.PR↗
arXiv2015

Approximate Normality of High-Energy Hyperspherical Eigenfunctions

The Berry heuristic has been a long standing \emph{ansatz} about the high energy (i.e. large eigenvalues) behaviour of eigenfunctions (see Berry 1977). Roughly speaking, it states that under some generic boundary conditions, these eigenfunctions exhibit Gaussian behaviour when the eigenvalues grow to infinity. Our aim in this paper is to make this statement quantitative and to establish some rigorous bounds on the distance to Gaussianity, focussing on the hyperspherical case (i.e., for eigenfunctions of the Laplace-Beltrami operator on the normalized $d$-dimensional sphere - also known as spherical harmonics). Some applications to non-Gaussian models are also discussed.

math.PR↗
arXiv2014

On the distribution of the critical values of random spherical harmonics

We study the limiting distribution of critical points and extrema of random spherical harmonics, in the high energy limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions for the variances and we show that the empirical measures in the high-energy limits converge weakly to their expected values. Our arguments require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed.

math-ph↗
arXiv2014

A Simple Proposal for Radial 3D Needlets

We present here a simple construction of a wavelet system for the three-dimensional ball, which we label \emph{Radial 3D Needlets}. The construction envisages a data collection environment where an observer located at the centre of the ball is surrounded by concentric spheres with the same pixelization at different radial distances, for any given resolution. The system is then obtained by weighting the projector operator built on the corresponding set of eigenfunctions, and performing a discretization step which turns out to be computationally very convenient. The resulting wavelets can be shown to have very good localization properties in the real and harmonic domain; their implementation is computationally very convenient, and they allow for exact reconstruction as they form a tight frame systems. Our theoretical results are supported by an extensive numerical analysis.

astro-ph.IM↗
arXiv2014

Gaussian approximations of nonlinear statistics on the sphere

We show how it is possible to assess the rate of convergence in the Gaussian approximation of triangular arrays of $U$-statistics, built from wavelets coefficients evaluated on a homogeneous spherical Poisson field of arbitrary dimension. For this purpose, we exploit the Stein-Malliavin approach introduced in the seminal paper by Peccati, Solé, Taqqu and Utzet (2011); we focus in particular on statistical applications covering evaluation of variance in non-parametric density estimation and Sobolev tests for uniformity.

math.PR↗
arXiv2014

The Needlet CMB Trispesctrum

We propose a computationally feasible estimator for the needlet trispectrum, which develops earlier work on the bispectrum by Donzelli et al. (2012). Our proposal seems to enjoy a number of useful properties, in particular a) the construction exploits the localization properties of the needlet system, and hence it automatically handles masked regions; b) the procedure incorporates a quadratic correction term to correct for the presence of instrumental noise and sky-cuts; c) it is possible to provide analytic results on its statistical properties, which can serve as a guidance for simulations. The needlet trispectrum we present here provides the natural building blocks for the efficient estimation of nonlinearity parameters on CMB data, and in particular for the third order constants $g_{NL}$ and $τ_{NL}$.

astro-ph.CO↗
arXiv2014

Cosmological Applications of the Gaussian Kinematic Formula

The Gaussian Kinematic Formula (GKF, see Adler and Taylor (2007,2011)) is an extremely powerful tool allowing for explicit analytic predictions of expected values of Minkowski functionals under realistic experimental conditions for cosmological data collections. In this paper, we implement Minkowski functionals on multipoles and needlet components of CMB fields, thus allowing a better control of cosmic variance and extraction of information on both harmonic and real domains; we then exploit the GKF to provide their expected values on spherical maps, in the presence of arbitrary sky masks, and under nonGaussian circumstances. All our results are validated by numerical experiments, which show a perfect agreement between theoretical predictions and Monte Carlo simulations.

astro-ph.IM↗
arXiv2014

Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on ${\mathbb{S}}^{d}$

We investigate Stein-Malliavin approximations for nonlinear functionals of geometric interest of Gaussian random eigenfunctions on the unit $d$ -dimensional sphere ${\mathbb{S}}^{d},$ $d\geq 2.$ All our results are established in the high energy limit, i.e. for eigenfunctions corresponding to growing eigenvalues. More precisely, we provide an asymptotic analysis for the variance of random eigenfunctions, and also establish rates of convergence for various probability metrics for Hermite subordinated processes, arbitrary polynomials of finite order and square integral nonlinear transforms; the latter, for instance, allows to prove a quantitative Central Limit Theorem for the excursion area. Some related issues were already considered in the literature for the $2$-dimensional case ${\mathbb{S}}^{2}$; our results are new or improve the existing bounds even for this special case. Proofs are based on the asymptotic analysis of moments of all order for Gegenbauer polynomials, and make extensive use of the recent literature on so-called fourth-moment theorems by Nourdin and Peccati.

math.PR↗
arXiv2014

A Note on Global Suprema of Band-Limited Spherical Random Functions

In this note, we investigate the behaviour of suprema for band-limited spherical random fields. We prove upper and lower bound for the expected values of these suprema, by means of metric entropy arguments and discrete approximations; we then exploit the Borell-TIS inequality to establish almost sure upper and lower bounds for their fluctuations. Band limited functions can be viewed as restrictions on the sphere of random polynomials with increasing degrees, and our results show that fluctuations scale as the square root of the logarithm of these degrees.

math.PR↗
arXiv2014

Gaussian semiparametric estimates on the unit sphere

We study the weak convergence (in the high-frequency limit) of the parameter estimators of power spectrum coefficients associated with Gaussian, spherical and isotropic random fields. In particular, we introduce a Whittle-type approximate maximum likelihood estimator and we investigate its asympotic weak consistency and Gaussianity, in both parametric and semiparametric cases.

math.ST↗
arXiv2014

Sparse Inpainting and Isotropy

Sparse inpainting techniques are gaining in popularity as a tool for cosmological data analysis, in particular for handling data which present masked regions and missing observations. We investigate here the relationship between sparse inpainting techniques using the spherical harmonic basis as a dictionary and the isotropy properties of cosmological maps, as for instance those arising from cosmic microwave background (CMB) experiments. In particular, we investigate the possibility that inpainted maps may exhibit anisotropies in the behaviour of higher-order angular polyspectra. We provide analytic computations and simulations of inpainted maps for a Gaussian isotropic model of CMB data, suggesting that the resulting angular trispectrum may exhibit small but non-negligible deviations from isotropy.

astro-ph.CO↗
arXiv2013

The Stochastic Properties of $\ell^1$-Regularized Spherical Gaussian Fields

Convex regularization techniques are now widespread tools for solving inverse problems in a variety of different frameworks. In some cases, the functions to be reconstructed are naturally viewed as realizations from random processes; an important question is thus whether such regularization techniques preserve the properties of the underlying probability measures. We focus here on a case which has produced a very lively debate in the cosmological literature, namely Gaussian and isotropic spherical random fields, and we prove that Gaussianity and isotropy are not conserved in general under convex regularization over a Fourier dictionary, such as the orthonormal system of spherical harmonics.

math.PR↗