Bridging between Soft and Hard Thresholding by Scaling

In this article, we developed and analyzed a thresholding method in which soft estimators are independently expanded by empirical scaling values. The values have common hyper-parameter that is an order of expansion ideal value achieves hard thresholding. We simply call estimator scaled estimator. general includes the non-negative garrote as special cases gives another derivation adaptive LASSO. then derived degree freedom means Stein's unbiased risk estimate found it decomposed into reminder connecting to meaning, natural bridge between methods. Since represents over-fitting, result implies there two sources over-fitting first source originated from determined number un-removed coefficients measure over-fitting. second particular case referring known for that, sparse, large sample non-parametric setting, largely coefficient estimates whose true zeros has influence on when threshold levels around noise those estimates. simple numerical example, these theoretical implications well explained behavior freedom. Moreover, based results here some facts, behaviors risks soft,