A polynomial algorithm for best-subset selection problem

. 2020 Dec 16;117(52):33117–33123. doi: 10.1073/pnas.2014241117

*Algorithm 2:* Splicing ( $β$ , $d$ , $A$ , $I$ , $k_{m a x}$ , $τ_{s}$ ).
1) Input: $β$ , $d$ , $A$ , $I$ , $k_{m a x}$ , and $τ_{s}$ .
2) Initialize $L_{0} = L = \frac{1}{2 n} ‖ y - X {β ‖}_{2}^{2}$ , and set
$ξ_{j} = \frac{X_{j}^{⊤} X_{j}}{2 n} {(β_{j})}^{2}, ζ_{j} = \frac{X_{j}^{⊤} X_{j}}{2 n} {(\frac{d_{j}}{X_{j}^{⊤} X_{j} / n})}^{2}, j = 1, \dots, p$ .
3) For $k = 1,2, \dots, k_{m a x}$ , do
$A_{k} = {j \in A : \sum_{i \in A} I (ξ_{j} \geq ξ_{i}) \leq k}$ ,
$I_{k} = {j \in I : \sum_{i \in I} I (ζ_{j} \leq ζ_{i}) \leq k} .$
Let ${\tilde{A}}_{k} = (A \ A_{k}) \cup I_{k}$ , ${\tilde{I}}_{k} = (I \ I_{k}) \cup A_{k}$ and solve
${\tilde{β}}_{{\tilde{A}}_{k}} = {(X_{{\tilde{A}}_{k}}^{⊤} X_{{\tilde{A}}_{k}})}^{- 1} X_{{\tilde{A}}_{k}}^{⊤} y, {\tilde{β}}_{{\tilde{I}}_{k}} = 0,$
$\tilde{d} = X^{⊤} (y - X \tilde{β}) / n, L_{n} (\tilde{β}) = \frac{1}{2 n} ‖ y - X {\tilde{β} ‖}_{2}^{2} .$
If $L > L_{n} (\tilde{β})$ , then
$(\hat{β}, \hat{d}, \hat{A}, \hat{I}) = (\tilde{β}, \tilde{d}, {\tilde{A}}_{k}, {\tilde{I}}_{k})$ ,
$L = L_{n} (\tilde{β}) .$
End for
4) If $L_{0} - L < τ_{s}$ , then $(\hat{β}, \hat{d}, \hat{A}, \hat{I}) = (β, d, A, I)$ .
5) Output ( $\hat{β}, \hat{d}, \hat{A}, \hat{I}$ ).