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

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.