P. Baldi, G. Kerkyacharian, D. Marinucci, D. Picard
This paper is concerned with density estimation of directional data on the sphere. We introduce a procedure based on thresholding on a new type of spherical wavelets called {\it needlets}. We establish a minimax result and prove its optimality. We are motivated by astrophysical applications, in particular in connection with the analysis of ultra high energy cosmic rays.
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The problem of estimating a probability density function f on the (d-1)-dimensional unit sphere S^{d-1} from directional data using the needlet frame is considered. It is shown that the decay of needlet coefficients supported near a point of a function f depends only on local Hölder continuity properties of f at x. This is then used to show that the thresholded needlet estimator introduced in Baldi, Kerkyacharian, Marinucci and Picard adapts to the local regularity properties of f. Moreover an adaptive confidence interval for f based on the thresholded needlet estimator is proposed, which is a…
Adaptive thresholding for wavelet-based nonparametric heteroskedastic variance estimation on the sphere
研究单位球面上空间变化的异方差函数非参数估计问题，提出基于局部球面框架、结合硬阈值协议和样本分割方案的全数据驱动多分辨率估计器，证明其在标准损失函数下达到极小极大最优收敛率，有空间适应性，在多领域有潜在应用。
This work is concerned with the study of asymptotic properties of nonparametric density estimates in the framework of circular data. The estimation procedure here applied is based on wavelet thresholding methods: the wavelets used are the so-called Mexican needlets, which describe a nearly-tight frame on the circle. We study the asymptotic behaviour of the $L^{2}$-risk function for these estimates, in particular its adaptivity, proving that its rate of convergence is nearly optimal.
In the random coefficients binary choice model, a binary variable equals 1 iff an index $X^\topβ$ is positive.The vectors $X$ and $β$ are independent and belong to the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$.We prove lower bounds on the minimax risk for estimation of the density $f\_β$ over Besov bodies where the loss is a power of the $L^p(\mathbb{S}^{d-1})$ norm for $1\le p\le \infty$. We show that a hard thresholding estimator based on a needlet expansion with data-driven thresholds achieves these lower bounds up to logarithmic factors.
We focus on the nonparametric density estimation problem with directional data. We propose a new rule for bandwidth selection for kernel density estimation. Our procedure is automatic, fully data-driven and adaptive to the smoothness degree of the density. We obtain an oracle inequality and optimal rates of convergence for the L2 error. Our theoretical results are illustrated with simulations.
Adaptive estimation of Sobolev-type energy functionals on the sphere
本文提出了一种基于球面需要量框架的自适应估计方法，用于估计球面上Sobolev型能量泛函，实现了在未知光滑度下的最优minimax速率。
Nonparametric needlet estimation for partial derivatives of a probability density function on the $d$-torus
本文提出基于环面needlet的局部阈值估计方法，用于估计$d$维环面上方向数据密度函数的偏导数，并证明了其在Besov空间上的极小极大最优性。
Let $X_1,...,X_n$ be a random sample from some unknown probability density $f$ defined on a compact homogeneous manifold $\mathbf M$ of dimension $d \ge 1$. Consider a 'needlet frame' $\{ϕ_{j η}\}$ describing a localised projection onto the space of eigenfunctions of the Laplace operator on $\mathbf M$ with corresponding eigenvalues less than $2^{2j}$, as constructed in \cite{GP10}. We prove non-asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator $f_n(j)$ obtained from an empirical estimate of the needlet projection $\sum_ηϕ_{j η} \int f ϕ_{…
Efficient Inference after Directionally Stable Adaptive Experiments
本文提出方向稳定性条件，证明在自适应数据下估计量保持渐近正态性和半参数效率，并为LinUCB采样下的常规标量目标提供了首个半参数效率保证。
Adaptive procedures for directional false discovery rate control
本文研究多重检验中方向性错误风险，证明两种自适应FDR方法在增加符号声明且假设独立时可强控制FDR_dir，尤其适用于真实原假设很少的高功效场景。
We revisit, in an original and challenging perspective, the problem of testing the null hypothesis that the mode of a directional signal is equal to a given value. Motivated by a real data example where the signal is weak, we consider this problem under asymptotic scenarios for which the signal strength goes to zero at an arbitrary rate~$η_n$. Both under the null and the alternative, we focus on rotationally symmetric distributions. We show that, while they are asymptotically equivalent under fixed signal strength, the classical Wald and Watson tests exhibit very different (null and non-null) …
We consider spherical data $X_i$ noised by a random rotation $\varepsilon_i\in$ SO(3) so that only the sample $Z_i=\varepsilon_iX_i$, $i=1,\dots, N$ is observed. We define a nonparametric test procedure to distinguish $H_0:$ ''the density $f$ of $X_i$ is the uniform density $f_0$ on the sphere'' and $H_1:$ ''$\|f-f_0\|_2^2\geq \Cψ_N$ and $f$ is in a Sobolev space with smoothness $s$''. For a noise density $f_\varepsilon$ with smoothness index $ν$, we show that an adaptive procedure (i.e. $s$ is not assumed to be known) cannot have a faster rate of separation than $ψ_N^{ad}(s)=(N/\sqrt{\log\log…
The aim of this paper is to study the nonparametric regression estimators on the sphere built by the needlet block thresholding. The block thresholding procedure proposed here follows the method introduced by Hall, Kerkyacharian and Picard in [Hall, Kerkyacharian, Picard, (1998), (1999)], modified to exploit the properties of the spherical standard needlets. Therefore, we will investigate on their convergence rates, attaining their adaptive properties over the Besov balls. This work is strongly motivated by issues arising in Cosmology and Astrophysics, concerning in particular the analysis of …
This work is concerned with the study of the adaptivity properties of nonparametric regression estimators over the $d$-dimensional sphere within the global thresholding framework. The estimators are constructed by means of a form of spherical wavelets, the so-called needlets, which enjoy strong concentration properties in both harmonic and real domains. The author establishes the convergence rates of the $L^p$-risks of these estimators, focussing on their minimax properties and proving their optimality over a scale of nonparametric regularity function spaces, namely, the Besov spaces.
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 gravitatio…
Gimbal Regression: Orientation-Adaptive Local Linear Regression under Spatial Heterogeneity
本文提出Gimbal回归，通过方向权重和确定性保障，实现稳定且可审计的局部回归估计，提升数值稳定性。
Adaptive directional estimator of the density in R^d for independent and mixing sequences
本文提出一种基于经验特征函数Fourier反演的多元密度自适应方向估计器，适用于独立与混合序列，无需依赖平滑参数即可实现方向自适应并达到最优收敛速率。
Post-Learning Inference for Combinatorial Optimizers with High-Dimensional Sparse Contextual Information via Minimal Directional Perturbation
研究数据依赖组合优化器结构特性的学习后推理，针对高维上下文多项逻辑模型，提出基于非光滑最大差异收益统计量的扰动测试，证明相关有效性等。
Parallel Network Reconstruction with Multi-directional Regularization
本文提出PALMS分布式并行框架，通过节点划分与自适应多方向正则化实现大规模网络的高效重构，并在理论上证明一致性，实验验证其计算效率与重构精度。
Linear Convergence of the Subspace Constrained Mean Shift Algorithm: From Euclidean to Directional Data
本文证明了子空间约束均值漂移算法的线性收敛性，并将其推广到方向数据，建立了方向数据密度脊的稳定性定理。
Precision matrix is of significant importance in a wide range of applications in multivariate analysis. This paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting. Optimal rates of convergence are established for a range of matrix norm losses. A fully data driven estimator based on adaptive constrained $\ell_1$ minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces. The estimator, called ACLIME, is easy to implement and performs well numerically. A major step in establishing the minimax rate of c…
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