English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Pro

Presents the zaia usage as premium feature

ZAIA

Presents AI Discovery search usage as a premium feature

AI Discovery

Presents AI Discovery explain and summary usage as a premium feature

Content Analysis

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Presents the pdf analysis as premium feature

PDF Analysis

1500 Top-up

3000 Top-up

Zendy Basic

2500 Top-up

Zendy Unlimited

Presents the access of premium content as premium feature

Premium Content

In this article, we developed and analyzed a thresholding method in whichsoft thresholding estimators are independently expanded by empirical scalingvalues. The scaling values have a common hyper-parameter that is an order ofexpansion of an ideal scaling value that achieves hard thresholding. We simplycall this estimator a scaled soft thresholding estimator. The scaled softthresholding is a general method that includes the soft thresholding andnon-negative garrote as special cases and gives an another derivation ofadaptive LASSO. We then derived the degree of freedom of the scaled softthresholding by means of the Stein's unbiased risk estimate and found that itis decomposed into the degree of freedom of soft thresholding and the reminderconnecting to hard thresholding. In this meaning, the scaled soft thresholdinggives a natural bridge between soft and hard thresholding methods. Since thedegree of freedom represents the degree of over-fitting, this result impliesthat there are two sources of over-fitting in the scaled soft thresholding. Thefirst source originated from soft thresholding is determined by the number ofun-removed coefficients and is a natural measure of the degree of over-fitting.We analyzed the second source in a particular case of the scaled softthresholding by referring a known result for hard thresholding. We then foundthat, in a sparse, large sample and non-parametric setting, the second sourceis largely determined by coefficient estimates whose true values are zeros andhas an influence on over-fitting when threshold levels are around noise levelsin those coefficient estimates. In a simple numerical example, thesetheoretical implications has well explained the behavior of the degree offreedom. Moreover, based on the results here and some known facts, we explainedthe behaviors of risks of soft, hard and scaled soft thresholding methods.

Bridging between soft and hard thresholding by scaling