Algorithm 1: BESS.Fix($s$): Best-Subset Selection with a given support size $s$.
1) Input: $\mathit{X}$, $\mathit{y}$, a positive integer $k_{max}$, and a threshold ${\tau }_s$.
2) Initialize ${\mathcal{A}}^0=\left\{j:{\sum }_{i=1}^p\mathrm{I}\left(|\frac{{\mathit{X}}_j^{\top }\mathit{y}}{\sqrt{{\mathit{X}}_j^{\top }{\mathit{X}}_j}}\right.|\le |\frac{{\mathit{X}}_i^{\top }\mathit{y}}{\sqrt{{\mathit{X}}_i^{\top }{\mathit{X}}_i}}\le s\right\},{\mathcal{I}}^0={\left({\mathcal{A}}^0\right)}^c$,
and (${\mathit{\beta }}^0$, ${\mathit{d}}^0$):
${\mathit{\beta }}_{{\mathcal{I}}^0}^0=0$,
${\mathit{d}}_{{\mathcal{A}}^0}^0=0$,
${\mathit{\beta }}_{{\mathcal{A}}^0}^0={\left({\mathit{X}}_{{\mathcal{A}}^0}^{\top }{\mathit{X}}_{{\mathcal{A}}^0}\right)}^{-1}{\mathit{X}}_{{\mathcal{A}}^0}^{\top }\mathit{y}$,
${\mathit{d}}_{{\mathcal{I}}^0}^0={\mathit{X}}_{{\mathcal{I}}^0}^{\top }\left(\mathit{y}-\mathit{X}{\mathit{\beta }}^0\right)/n.$
3) For $m=\mathrm{0,1},\dots \hspace{0.17em}$, do
$$\left({\mathit{\beta }}^{m+1},{\mathit{d}}^{m+1},{\mathcal{A}}^{m+1},{\mathcal{I}}^{m+1}\right)=\text{Splicing}({\mathit{\beta }}^m,{\mathit{d}}^m,{\mathcal{A}}^m,{\mathcal{I}}^m,k_{max},{\tau }_s).$$
If $\left({\mathcal{A}}^{m+1},{\mathcal{I}}^{m+1}\right)=\left({\mathcal{A}}^m,{\mathcal{I}}^m\right)$, then stop
end for
4) Output $\left(\widehat{\mathit{\beta }},\widehat{\mathit{d}},\widehat{\mathcal{A}},\widehat{\mathcal{I}}\right)=\left({\mathit{\beta }}^{m+1},{\mathit{d}}^{m+1}{\mathcal{A}}^{m+1},{\mathcal{I}}^{m+1}\right)$.

Note that splicing size k is an important parameter in splicing. Typically, we can try all possible values of k≤s.