Adaptive global thresholding on the sphere

Durastanti, Claudio

doi:10.1016/j.jmva.2016.07.009

Cited by 10 publications

(13 citation statements)

References 47 publications

(155 reference statements)

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“…Moreover, this kind of results can be applied to solve problems on the hypersphere as those investigated in [Mar06,Mar08] for the two-dimensional case; we believe they could be useful in order to study also other statistical issues on S d , a topic which has recently received some attention (see e.g. [Dur16]).…”

Section: Resultsmentioning

confidence: 90%

The Defect of Random Hyperspherical Harmonics

Rossi

2018

J Theor Probab

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Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-sphere (d ≥ 2). We investigate the distribution of their defect i.e., the difference between the measure of positive and negative regions. Marinucci and Wigman studied the two-dimensional case giving the asymptotic variance [MW11] and a Central Limit Theorem [MW14], both in the high-energy limit.Our main results concern asymptotics for the defect variance and quantitative CLTs in Wasserstein distance, in any dimension. The proofs are based on Wiener-Itô chaos expansions for the defect, a careful use of asymptotic results for all order moments of Gegenbauer polynomials and Stein-Malliavin approximation techniques by Nourdin and Peccati [NP09,NP12]. Our argument requires some novel technical results of independent interest that involve integrals of the product of three hyperspherical harmonics.

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Section: Resultsmentioning

confidence: 90%

The Defect of Random Hyperspherical Harmonics

Rossi

2018

J Theor Probab

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“…This result first appeared in [Dur16] for needlets over the d-dimensional sphere S d . The original proof can be easily extended to the compact manifold framework and, therefore, is omitted here for the sake of brevity.…”

Section: Main Results: De-poissonized Casementioning

confidence: 82%

“…Among several techniques developed for this purpose, we focus on the so-called global thresholding needlet method, cf. [Dur16]. In this case, an estimator for f is given by…”

Section: Main Results: De-poissonized Casementioning

confidence: 99%

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On high-frequency limits of $U$-statistics in Besov spaces over compact manifolds

Bourguin¹,

Durastanti²

2017

Illinois J. Math.

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In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based U -statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered here are the so-called needlets, characterized by strong concentration properties and by an exact reconstruction formula. Furthermore, we consider Poisson point processes over the manifold such that the density function associated to its control measure lives in a Besov space. The main findings of this paper include new rates of convergence that depend strongly on the degree of regularity of the control measure of the underlying Poisson point process, providing a refined understanding of the connection between regularity and speed of convergence in this framework.2010 Mathematics Subject Classification. 60B05,60F05,60G57,62E20.

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“…They obtained minimax rates of convergence for B s q,r Besov spaces, L p -loss and sup-norm loss up to a logarithmic factor. This deep approach was continued in [12] but in regression case. Moreover the rates are without the logarithmic factor.…”

Section: Introductionmentioning

confidence: 99%

Multiresolution analysis and adaptive estimation on a sphere using stereographic wavelets

2019

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We construct an adaptive estimator of a density function on d dimensional unit sphere S d (d ≥ 2), using a new type of spherical frames. The frames, or as we call them, stereografic wavelets are obtained by transforming a wavelet system, namely Daubechies, using some stereographic operators. We prove that our estimator achieves an optimal rate of convergence on some Besov type class of functions by adapting to unknown smoothness. Our new construction of stereografic wavelet system gives us a multiresolution approximation of L 2 (S d ) which can be used in many approximation and estimation problems. In this paper we also demonstrate how to implement the density estimator in S 2 and we present a finite sample behavior of that estimator in a numerical experiment. n i=1

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Adaptive global thresholding on the sphere

Cited by 10 publications

References 47 publications

The Defect of Random Hyperspherical Harmonics

The Defect of Random Hyperspherical Harmonics

On high-frequency limits of $U$-statistics in Besov spaces over compact manifolds

Multiresolution analysis and adaptive estimation on a sphere using stereographic wavelets

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