Algorithm 1.

Sequential conditional independent tuples (multiple knockoffs)

$j=1$

while $j\le p$ do

Sample ${\tilde{\mathbf{x}}}_j^1,\dots ,{\tilde{\mathbf{x}}}_j^M$ independently from $ℒ\left({\mathbf{x}}_j\mid {\mathbf{x}}_{-j},{\tilde{\mathbf{x}}}_{1:j-1}^1,\dots ,{\tilde{\mathbf{x}}}_{1:j-1}^M\right)$

$j=j+1$

End

where $ℒ\left({\mathbf{x}}_j\mid {\mathbf{x}}_{-j},{\tilde{\mathbf{x}}}_{1:j-1}^1,\cdots ,{\tilde{\mathbf{x}}}_{1:j-1}^M\right)$ is the conditional distribution of ${\mathbf{x}}_j$ given ${\mathbf{x}}_{-j},{\tilde{\mathbf{x}}}_{1:j-1}^1,\cdots ,{\tilde{\mathbf{x}}}_{1:j-1}^M$ where $-j$ indicates the variable ${\mathbf{x}}_j$ is excluded and $M$ is the total number of knockoffs.