Adaptive nonparametric regression on spin fiber bundles

Abstract

The construction of adaptive nonparametric procedures by means of wavelet thresholding techniques is now a classical topic in modern mathematical statistics. In this paper, we extend this framework to the analysis of nonparametric regression on sections of spin fiber bundles defined on the sphere. This can be viewed as a regression problem where the function to be estimated takes as its values algebraic curves (for instance, ellipses) rather than scalars, as usual. The problem is motivated by many important astrophysical applications, concerning for instance the analysis of the weak gravitational lensing effect, i.e. the distortion effect of gravity on the images of distant galaxies. We propose a thresholding procedure based upon the (mixed) spin needlets construction recently advocated by Marinucci (2008,2010) and ), and we investigate their rates of convergence and their adaptive properties over spin Besov balls.

Adaptive global thresholding on the sphere

“…We note again that the method established here can be viewed as an extension of global thresholding techniques to the needlet regression function estimation. In this sense, our results are strongly related to those presented in [1,10,30], as discussed in Section 1.3.…”

“…Proof of Theorem 1.1. Following, for instance, [1,7,9,10,22,30] and as mentioned in Section 3.3, the L p -risk E( f n − f p L p (S d ) ) can be decomposed as the sum of a stochastic and a bias term. More specifically,…”

“…Following Durastanti et al [10], the labels A and U denote the regions where Θ j (p) is larger and smaller than the threshold B dj n −p/2 respectively, whereas a and u refer to the regions where the deterministic Θ j (p) are above and under a new threshold, given by 2 −1 B dj n −p/2 for a and 2B dj n −p/2 for u. The decay of Aa and U u depends on the properties of Besov spaces, while the bounds on Au and U a depend on the probabilistic inequalities concerning β j,k and Θ j (p), given in Propositions 3.1, 3.3 and 3.4.…”

“…The block thresholding framework was investigated in Durastanti [9]. Furthermore, the adaptivity of nonparametric regression estimators of spin function was studied in Durastanti et al [10]. In this case, the regression function takes as its values algebraical curves lying on the tangent plane for each point on S 2 and the wavelets used are the so-called spin (pure and mixed) needlets; see Geller and Marinucci [16,17].…”

“…, ε n to be null-mean independent random variables with finite absolute pth moment. However, we include the notion of sub-Gaussianity in order to be consistent with the existing literature; see [10]. Furthermore, sub-Gaussianity involves a wide class of random variables, including Gaussian and bounded random variables and, in general, all the random variables such that their moment generating function has a an upper bound in terms of the moment generating function of a centered Gaussian random variable of variance a.…”