Basis Pursuit ADMM

news2024/11/27 1:20:51

c笔记 ref. distr_opt_stat_learning_admm.html

Basis pursuit is the equality-constrained $\mathit{l}_{1}$ minimization problem

In ADMM form, basis pursuit can be written as
$\text{minimize} \, \, f(x) + \left \| z \right \|_1 \\ \text{subject to} \,\, x-z = 0$

The ADMM algorithm is then

$x^{k+1} :=\Pi\left(z^k-u^k\right) \\ z^{k+1} :=S_{1 / \rho}\left(x^{k+1}+u^k\right) \\ u^{k+1} :=u^k+x^{k+1}-z^{k+1},$

The x-update, which involves solving a linearly-constrained minimum Euclidean norm problem, can be written explicitly as

$x^{k+1}:=\left(I-A^T\left(A A^T\right)^{-1} A\right)\left(z^k-u^k\right)+A^T\left(A A^T\right)^{-1} b$

4.4.3 Soft Thresholding

where the soft thresholding operator S is defined as
$S_\kappa(a)= \begin{cases}a-\kappa & a>\kappa \\ 0 & |a| \leq \kappa \\ a+\kappa & a<-\kappa\end{cases}$

$S_\kappa(a)=(1-\kappa /|a|)+a$

the comments on efficient computation from §4.2 apply

We can interpret ADMM for basis pursuit as reducing the solution of a least $\mathit{l}_{1}$ norm problem to solving a sequence of minimum Euclidean norm problem

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