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Degrees of freedom in submodular regularization: A computational perspective of Stein’s unbiased risk estimate

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  • Minami, Kentaro

Abstract

Degrees of freedom is a covariance penalty related to penalized model selection procedures such as Mallows’ Cp and AIC. We study the degrees of freedom of two polyhedral convex regularization classes defined through submodular functions called the Lovász extension regularization and submodular norm regularization. It has been pointed out that submodular regularization contains many existing penalties that induce structural sparsity. In this paper, we show that the degrees of freedom of submodular regularization estimators can be represented in terms of partitions induced by the estimators. Our formula does not depend on the choice of the design matrix and the penalty function. Moreover, if the design matrix has full column rank, calculating an unbiased estimator of the degrees of freedom requires an additional computational cost of only O(plogp) after a solution for the estimator is obtained, where p is the dimension of the parameter. Existing results for some regularization and projection type estimators, such as the lasso, the fused lasso, and the isotonic regression, are also recovered.

Suggested Citation

  • Minami, Kentaro, 2020. "Degrees of freedom in submodular regularization: A computational perspective of Stein’s unbiased risk estimate," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    • Handle: RePEc:eee:jmvana:v:175:y:2020:i:c:s0047259x18305906
    DOI: 10.1016/j.jmva.2019.104546
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    References listed on IDEAS

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    1. Kato, Kengo, 2009. "On the degrees of freedom in shrinkage estimation," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1338-1352, August.
    2. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
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    4. Ryan J. Tibshirani & Saharon Rosset, 2019. "Excess Optimism: How Biased is the Apparent Error of an Estimator Tuned by SURE?," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(526), pages 697-712, April.
    5. Howard D. Bondell & Brian J. Reich, 2008. "Simultaneous Regression Shrinkage, Variable Selection, and Supervised Clustering of Predictors with OSCAR," Biometrics, The International Biometric Society, vol. 64(1), pages 115-123, March.
    6. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
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    Cited by:

    1. Guillaume Allaire Pouliot & Zhen Xie, 2022. "Degrees of Freedom and Information Criteria for the Synthetic Control Method," Papers 2207.02943, arXiv.org.

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