A Scaling and Non-Negative Garrote in Soft-Thresholding

Katsuyuki HAGIWARA  

IEICE TRANSACTIONS on Information and Systems   Vol.E100-D   No.11   pp.2702-2710
Publication Date: 2017/11/01
Publicized: 2017/07/27
Online ISSN: 1745-1361
DOI: 10.1587/transinf.2016EDP7365
Type of Manuscript: PAPER
Category: Artificial Intelligence, Data Mining
soft-thresholding,  SURE,  non-negative garrote,  scaling,  wavelet denoising,  

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Soft-thresholding is a sparse modeling method typically applied to wavelet denoising in statistical signal processing. It is also important in machine learning since it is an essential nature of the well-known LASSO (Least Absolute Shrinkage and Selection Operator). It is known that soft-thresholding, thus, LASSO suffers from a problem of dilemma between sparsity and generalization. This is caused by excessive shrinkage at a sparse representation. There are several methods for improving this problem in the field of signal processing and machine learning. In this paper, we considered to extend and analyze a method of scaling of soft-thresholding estimators. In a setting of non-parametric orthogonal regression problem including discrete wavelet transform, we introduced component-wise and data-dependent scaling that is indeed identical to non-negative garrote. We here considered a case where a parameter value of soft-thresholding is chosen from absolute values of the least squares estimates, by which the model selection problem reduces to the determination of the number of non-zero coefficient estimates. In this case, we firstly derived a risk and construct SURE (Stein's unbiased risk estimator) that can be used for determining the number of non-zero coefficient estimates. We also analyzed some properties of the risk curve and found that our scaling method with the derived SURE is possible to yield a model with low risk and high sparsity compared to a naive soft-thresholding method with SURE. This theoretical speculation was verified by a simple numerical experiment of wavelet denoising.