Smooth Blockwise Iterative Thresholding: A Smooth Fixed Point Estimator Based on the Likelihood’s Block Gradient

Sylvain Sardy

Journal of the American Statistical Association, 2012, vol. 107, issue 498, 800-813

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 λ and ν, 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 λ and ν, 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 derives a universal threshold, an information criterion, and an oracle inequality in this canonical setting.

Date: 2012
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DOI: 10.1080/01621459.2012.664527

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