Beyond Thresholding: Analysis and Improvements for Deterministic Parameter Estimation

Coherence-Based Performance Guarantees for Estimating a Sparse Vector Under Random Noise

We consider the problem of estimating a deterministic sparse vector x from underdetermined measurements Ax+w, where w represents white Gaussian noise and A is a given deterministic dictionary. We analyze the performance of three sparse estimation algorithms: basis pursuit denoising (BPDN), orthogonal matching pursuit (OMP), and thresholding. These algorithms are shown to achieve near-oracle performance with high probability, assuming that x is sufficiently sparse. Our results are non-asymptotic and are based only on the coherence of A, so that they are applicable to arbitrary dictionaries. Dif…

基于抖动差分逐步下降阈值的鲁棒模型阶数选择

Robust Model Order Selection via Dithered Differential Step-Down Thresholding

本文针对平稳噪声中模型阶数选择问题，提出带随机网格抖动的差分逐步下降阈值算法，可实现渐近精确阶数恢复且鲁棒性强。

Hyperanalytic Denoising

A new thresholding strategy for the estimation of a deterministic image immersed in noise is introduced. The threshold is combined with a wavelet decomposition, where the wavelet coefficient of the image at any fixed value of the decomposition index is estimated, via thresholding the observed coefficient depending on the value of both the magnitude of the observed coefficient as well as the magnitudes of coefficients of a set of additional images calculated from the observed image. The additional set of images is chosen so that the wavelet transforms of the full set of images have suitable det…

一类具有去偏和阈值化的高维线性回归方法

A unified spectral analysis of high-dimensional linear regression: debiasing and thresholding

本文提出统一正则化框架，涵盖传统及七种新回归方法，并引入去偏阈值化技术以促进稀疏特征选择，理论上建立相合性与高斯逼近，数值模拟显示其有限样本性能优越。

Nonasymptotic estimation and support recovery for high dimensional sparse covariance matrices

We propose a general framework for nonasymptotic covariance matrix estimation making use of concentration inequality-based confidence sets. We specify this framework for the estimation of large sparse covariance matrices through incorporation of past thresholding estimators with key emphasis on support recovery. This technique goes beyond past results for thresholding estimators by allowing for a wide range of distributional assumptions beyond merely sub-Gaussian tails. This methodology can furthermore be adapted to a wide range of other estimators and settings. The usage of nonasymptotic dime…

Empirical Bayes Estimators for High-Dimensional Sparse Vectors

The problem of estimating a high-dimensional sparse vector $\boldsymbolθ \in \mathbb{R}^n$ from an observation in i.i.d. Gaussian noise is considered. The performance is measured using squared-error loss. An empirical Bayes shrinkage estimator, derived using a Bernoulli-Gaussian prior, is analyzed and compared with the well-known soft-thresholding estimator. We obtain concentration inequalities for the Stein's unbiased risk estimate and the loss function of both estimators. The results show that for large $n$, both the risk estimate and the loss function concentrate on deterministic values clo…