Smooth blockwise iterative thresholding: a smooth fixed point estimator based on the likelihood's block gradient

TL;DR

This paper introduces SBITE, a smooth fixed point estimator combining ridge and lasso features, with a focus on Gaussian regression and applications to gravitational wave detection, offering improved parameter selection and risk estimation.

Contribution

It proposes SBITE, a novel smooth blockwise iterative thresholding method with a smooth risk estimate, enhancing variable selection and model tuning in Gaussian regression.

Findings

SBITE is uniquely defined and has a smooth Stein unbiased risk estimate.

Simulation shows SBITE's favorable predictive and oracle properties.

Application to gravitational wave detection demonstrates practical effectiveness.

Abstract

The proposed smooth blockwise iterative thresholding estimator (SBITE) is a model selection technique defined as a fixed point reached by iterating a likelihood gradient-based thresholding function. The smooth James-Stein thresholding function has two regularization parameters $\lambda$ and $\nu$, and a smoothness parameter $s$. It enjoys smoothness like ridge regression and selects variables like lasso. Focusing on Gaussian regression, we show that SBITE is uniquely defined, and that its Stein unbiased risk estimate is a smooth function of $\lambda$ and $\nu$, for better selection of the two regularization parameters. We perform a Monte-Carlo simulation to investigate the predictive and oracle properties of this smooth version of adaptive lasso. The motivation is a gravitational wave burst detection problem from several concomitant time series. A nonparametric wavelet-based estimator is developed to combine information from all captors by block-thresholding multiresolution coefficients. We study how the smoothness parameter $s$ tempers the erraticity of the risk estimate, and derive a universal threshold, an information criterion and an oracle inequality in this canonical setting.