Algorithm 1.

Efficient algorithm for constructing SFN_simplex

Input: Normalized data $\mathit{X}\in {ℝ}^{n\times p}$, and parameter λG.
Output : $\mathit{S}\in {ℝ}^{p\times p}$
1:	Initialize c(0), ${\mathit{\alpha }}_i^{(0)}$
2:	for each region i (i = 1, ..., p) do
3:	Set k = 0
4:	While no convergence do
5:	k = k + 1
6:	${\mathit{\beta }}_i^{(k)}={\min }_{{\mathit{\beta }}_i}\frac{1}{2}{\Vert {\mathit{\beta }}_i-{\mathit{\alpha }}_i^{(k-1)}\Vert }_2^{2}s.t.{\mathit{\beta }}_i\ge 0,{\mathit{\beta }}_i^T\mathbf{1}=1$,
7:	${\mathit{\alpha }}_i^{(k)}={\mathit{z}}_i^{(k)}-t\nabla f\left({\mathit{z}}_i^{(k)}\right)$, $\mathrm{where}\nabla f\left({\mathit{\beta }}_i^{(k)}\right)={\mathit{X}}_{(:,\ne i)}^{\text{T}}{\mathit{X}}_{(:,\ne i)}{\mathit{\beta }}_i^{(k)}+{\lambda }_G{\mathit{L}}_{sn}{\mathit{\beta }}_i^{(k)}-{\mathit{X}}_{(:,\ne i)}{\mathit{X}}_{(:,i)}$
8:	${\mathit{z}}_i^{(k)}={\mathit{\beta }}_i^{(k)}+\frac{c^{(k)}-1}{c^{(k+1)}}({\mathit{\beta }}_i^{(k)}-{\mathit{\beta }}_i^{(k-1)})$
9:	$c^{(k)}=\frac{1+\sqrt{1+4c^{{(k-1)}^2}}}{2}$
10:	end
11:	$\widehat{{\mathit{\beta }}_{ı}}=\left[{\beta }_1,\dots ,{\beta }_{i-1},0,{\beta }_{i+1},\dots ,{\beta }_p\right]$
12:	end for
13:	$\mathit{S}=[\widehat{{\mathit{\beta }}_1},\dots ,\widehat{{\mathit{\beta }}_p}]$
14:	$\mathit{S}\mathit{F}{\mathit{N}}_{simplex}=\frac{\mathit{S}+{\mathit{S}}^{\text{T}}}{2}$