TY - CHAP

T1 - A Tutorial on

T2 - R Package for the Linearized Bregman Algorithm in High-Dimensional Statistics

AU - Xiong, Jiechao

AU - Ruan, Feng

AU - Yao, Yuan

PY - 2018

Y1 - 2018

N2 - The R package,Libra, stands for the LInearized BRegman Algorithm in high-dimensional statistics. The Linearized Bregman Algorithm is a simple iterative procedure which generates sparse regularization paths of model estimation. This algorithm was firstly proposed in applied mathematics for image restoration, and is particularly suitable for parallel implementation in large-scale problems. The limit of such an algorithm is a sparsity-restricted gradient descent flow, called the Inverse Scale Space, evolving along a parsimonious path of sparse models from the null model to overfitting ones. In sparse linear regression, the dynamics with early stopping regularization can provably meet the unbiased oracle estimator under nearly the same condition as LASSO, while the latter is biased. Despite its successful applications, proving the consistency of such dynamical algorithms remains largely open except for some recent progress on linear regression. In this tutorial, algorithmic implementations in the package are discussed for several widely used sparse models in statistics, including linear regression, logistic regression, and several graphical models (Gaussian, Ising, and Potts). Besides the simulation examples, various applications are demonstrated, with real-world datasets such as diabetes, publications of COPSS award winners, as well as social networks of two Chinese classic novels, Journey to the West and Dream of the Red Chamber.

AB - The R package,Libra, stands for the LInearized BRegman Algorithm in high-dimensional statistics. The Linearized Bregman Algorithm is a simple iterative procedure which generates sparse regularization paths of model estimation. This algorithm was firstly proposed in applied mathematics for image restoration, and is particularly suitable for parallel implementation in large-scale problems. The limit of such an algorithm is a sparsity-restricted gradient descent flow, called the Inverse Scale Space, evolving along a parsimonious path of sparse models from the null model to overfitting ones. In sparse linear regression, the dynamics with early stopping regularization can provably meet the unbiased oracle estimator under nearly the same condition as LASSO, while the latter is biased. Despite its successful applications, proving the consistency of such dynamical algorithms remains largely open except for some recent progress on linear regression. In this tutorial, algorithmic implementations in the package are discussed for several widely used sparse models in statistics, including linear regression, logistic regression, and several graphical models (Gaussian, Ising, and Potts). Besides the simulation examples, various applications are demonstrated, with real-world datasets such as diabetes, publications of COPSS award winners, as well as social networks of two Chinese classic novels, Journey to the West and Dream of the Red Chamber.

U2 - https://doi.org/10.1007/978-3-319-18284-1_17

DO - https://doi.org/10.1007/978-3-319-18284-1_17

M3 - Chapter

SN - 978-3319182834

T3 - Springer Handbooks of Computational Statistics

BT - Handbook of Big Data Analytics

PB - Springer

ER -