Optimal Spherical Deconvolution

This paper addresses the issue of optimal deconvolution density estimation on the 2-sphere. Indeed, by using the transitive group action of the rotation matrices on the 2-dimensional unit sphere, rotational errors can be introduced analogous to the Euclidean case. The resulting density turns out to be convolution in the Lie group sense and so the statistical problem is to recover the true underlying density. This recovery can be done by deconvolution; however, as in the Euclidean case, the difficulty of the deconvolution turns out to depend on the spectral properties of the rotational error distribution. This therefore leads us to define smooth and super-smooth classes and optimal rates of convergence are obtained for these smoothness classes.

[1]  Eugene P. Wigner,et al.  Special Functions. A Group Theoretic Approach , 1969 .

[2]  N. D. Mermin,et al.  Special Functions: A Group Theoretic Approach , 1970 .

[3]  K. Mardia,et al.  The von Mises–Fisher Matrix Distribution in Orientation Statistics , 1977 .

[4]  Ja-Yong Koo,et al.  Optimal Rates of Convergence for Nonparametric Statistical Inverse Problems , 1993 .

[5]  Harrie Hendriks,et al.  Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions , 1990 .

[6]  Jianqing Fan.  On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems , 1991 .

[7]  Related Topics,et al.  Nonparametric functional estimation and related topics , 1991 .

[8]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[9]  S. J. Patterson,et al.  HARMONIC ANALYSIS ON SYMMETRIC SPACES AND APPLICATIONS , 1990 .

[10]  F. Ruymgaart,et al.  Regularized Deconvolution on the Circle and the Sphere , 1991 .

[11]  Linda R. Eshleman,et al.  Exponential Fourier Densities on $SO( 3 )$ and Optimal Estimation and Detection for Rotational Processes , 1979 .

[12]  Sam Efromovich,et al.  Density Estimation for the Case of Supersmooth Measurement Error , 1997 .

[13]  Jianqing Fan,et al.  Global Behavior of Deconvolution Kernel Estimates , 1989 .

[14]  Jianqing Fan,et al.  Adaptively Local One-Dimensional Subproblems with Application to a Deconvolution Problem , 1993 .

[15]  Dennis M. Healy,et al.  An empirical Bayes approach to directional data and efficient computation on the sphere , 1996 .

[16]  Dennis M. Healy,et al.  Spherical Deconvolution , 1998 .

[17]  Tammo tom Dieck,et al.  Representations of Compact Lie Groups , 1985 .

[18]  A. Terras.  Harmonic Analysis on Symmetric Spaces and Applications I , 1985 .

[19]  J. Rosenthal.  Random Rotations: Characters and Random Walks on SO(N) , 1994 .

[20]  K. Maurin.  Representations of Compact Lie Groups , 1997 .

[21]  G. Wahba.  Spline Interpolation and Smoothing on the Sphere , 1981 .

[22]  Y. Yatracos.  A Lower Bound on the Error in Nonparametric Regression Type Problems , 1988 .

[23]  C. J. Stone,et al.  Optimal Rates of Convergence for Nonparametric Estimators , 1980 .

[24]  Linda R. Eshleman,et al.  Exponential Fourier densities on SO(3) and optimal estimation and detection for rotational processes , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[25]  Lucien Birgé.  Approximation dans les espaces métriques et théorie de l'estimation , 1983 .

[26]  Ja-Yong Koo,et al.  Log-density estimation in linear inverse problems , 1998 .