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Sparsity and smoothness via the fused lasso. (English) Zbl 1060.62049
Summary: The lasso penalizes a least squares regression by the sum of the absolute values $$(L_1$$-norm) of the coefficients. The form of this penalty encourages sparse solutions (with many coefficients equal to $$0)$$. We propose the ‘fused lasso’, a generalization that is designed for problems with features that can be ordered in some meaningful way. The fused lasso penalizes the $$L_1$$-norm of both the coefficients and their successive differences. Thus it encourages sparsity of the coefficients and also sparsity of their differences – i.e., local constancy of the coefficient profile. The fused lasso is especially useful when the number of features $$p$$ is much greater than $$N$$, the sample size. The technique is also extended to the ‘hinge’ loss function that underlies the support vector classifier. We illustrate the methods on examples from protein mass spectroscopy and gene expression data.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62P10 Applications of statistics to biology and medical sciences; meta analysis 62J05 Linear regression; mixed models
##### Software:
SQOPT; ElemStatLearn
Full Text:
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