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从领头变量中学习：LOL 高维回归方法AI 解读

Summary The paper investigates the estimation problem in a regression-type model. To be able to deal with potential high dimensions, we provide a procedure called LOL—for learning out of leaders—with no optimization step. LOL is an autodriven algorithm with two thresholding steps. A first adaptive thresholding helps to select leaders among the initial regressors to obtain a first reduction of dimensionality. Then a second thresholding is performed on the linear regression on…

局部化球面反卷积AI 解读

We provide a new algorithm for the treatment of the deconvolution problem on the sphere which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. We establish upper bounds for the behavior of our procedure for any $\mathbb {L}_{p}$ loss. It is important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover as well as to inhomogeneous smoothness. We…

Summary Deconvolution problems are naturally represented in the Fourier domain, whereas thresholding in wavelet bases is known to have broad adaptivity properties. We study a method which combines both fast Fourier and fast wavelet transforms and can recover a blurred function observed in white noise with O{n log (n)2} steps. In the periodic setting, the method applies to most deconvolution problems, including certain ‘boxcar’ kernels, which are important as a model of…

曲线点值估计的自适应置信区间AI 解读

We present a procedure associated with nonlinear wavelet methods that provides adaptive confidence intervals around $f (x_0)$, in either a white noise model or a regression setting. A suitable modification in the truncation rule for wavelets allows construction of confidence intervals that achieve optimal coverage accuracy up to a logarithmic factor. The procedure does not require knowledge of the regularity of the unknown function $f$; it is also efficient for functions…

核与小波曲线估计中的块阈值规则AI 解读

Motivated by recently developed threshold rules for wavelet estimators, we suggest threshold methods for general kernel density estimators, including those of classical Rosenblatt–Parzen type. Thresholding makes kernel methods competitive in terms of their adaptivity to a wide variety of aberrations in complex signals. It is argued that term-by-term thresholding does not always produce optimal performance, since individual coefficients cannot be estimated sufficiently…

基于小波阈值化的密度估计AI 解读

Density estimation is a commonly used test case for nonparametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients. Minimax rates of convergence are studied over a large range of Besov function classes $B_{\sigma pq}$ and for a range of global $L'_p$ error measures, $1 \leq p' < \infty$. A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a…

小波收缩：渐近理想之境？AI 解读

SUMMARY Much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects—curves, densities, spectral densities, images—from noisy data. A now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems. Unfortunately, the results have rarely moved into practice, for a variety of reasons—among them being similarity to known methods, computational intractability and lack of spatial…