Algorithm 1. Finding regularization parameters in griPEER

$$\begin{gathered} \begin{array}{cc}\hfill & \mathrm{Input}:\text{matrices:}Z,X\text{and}Q;\text{vector:}y;\text{initial point:}\stackrel{[0]}{\lambda }≔[\stackrel{[0]}{{\lambda }_Q},\stackrel{[0]}{{\lambda }_R}{]}^{\mathsf{T}};\text{stop criterion:}\delta >0;\hfill \\ \hfill & \text{function which defines the density:}\psi ;k≔1\hfill \\ \hfill & \mathbf{do}\hfill \\ \hfill & \\ 1.\text{define}\stackrel{[\mathrm{k}-1]}{\beta }\text{and}\stackrel{[\mathrm{k}-1]}{b}\text{by solving:}\hfill \\ \hfill & \\ \underset{\beta ,b}{\text{argmin}}\left\{-2{\sum }_{i=1}^n\left[y_i(X_i\beta +Z_{i}b)-\psi (X_i\beta +Z_{i}b)\right]+\stackrel{[\mathrm{k}-1]}{{\lambda }_Q}{b}^{T}Qb+\stackrel{[\mathrm{k}-1]}{{\lambda }_R}\Vert b{\Vert }_2^2\right\};\hfill \\ \hfill & \\ 2.\stackrel{[\mathrm{k}-1]}{\theta }≔X\stackrel{[\mathrm{k}-1]}{\beta }+Z\stackrel{[\mathrm{k}-1]}{b},\stackrel{[\mathrm{k}]}{W}≔\text{diag}\left([{\psi }^{″}(\stackrel{[\mathrm{k}-1]}{{\theta }_1}){]}^{-1},\dots ,[{\psi }^{″}(\stackrel{[\mathrm{k}-1]}{{\theta }_n}){]}^{-1}\right);\hfill \\ \hfill & \\ 3.\text{define}\stackrel{[\mathrm{k}]}{y}\text{by putting}\stackrel{[\mathrm{k}]}{y_i}≔[{\psi }^{″}(\stackrel{[\mathrm{k}-1]}{{\theta }_i}){]}^{-1}\cdot (y_i-{\psi }^{\prime }(\stackrel{[\mathrm{k}-1]}{{\theta }_i}))+\stackrel{[\mathrm{k}-1]}{{\theta }_i},\text{for}i=1,\dots ,n;\hfill \\ \hfill & \\ 4.\stackrel{[\mathrm{k}]}{\mathcal{P}}≔\mathrm{I}-X(X^{\mathsf{T}}\stackrel{[\mathrm{k}]}{W^{-1}}X{)}^{-1}{X}^{\mathsf{T}}\stackrel{[\mathrm{k}]}{W^{-1}};\hfill \\ \hfill & \\ 5.\stackrel{[\mathrm{k}]}{\tilde{y}}≔\stackrel{[\mathrm{k}]}{\mathcal{P}}\stackrel{[\mathrm{k}]}{y},\stackrel{[\mathrm{k}]}{X}≔\stackrel{[\mathrm{k}]}{\mathcal{P}}X,\stackrel{[\mathrm{k}]}{Z}≔\stackrel{[\mathrm{k}]}{\mathcal{P}}Z;\hfill \\ \hfill & \\ 6.\stackrel{[\mathrm{k}]}{\Omega }≔\stackrel{[\mathrm{k}]}{TZ}\stackrel{[\mathrm{k}]-1}{W}\stackrel{[\mathrm{k}]}{Z},\stackrel{[\mathrm{k}]}{q}≔\stackrel{[\mathrm{k}]\mathsf{T}}{Z}\stackrel{[\mathrm{k}]-1}{W}\stackrel{[\mathrm{k}]}{\tilde{y}};\hfill \\ \hfill & \\ 7.\stackrel{[\mathrm{k}]}{\lambda }≔\underset{\lambda \underset{¯}{\succ }0}{\text{argmin}}\left\{\text{ln}\mid \left({\lambda }_{Q}Q+{\lambda }_R{\mathrm{I}}_p+\stackrel{[\mathrm{k}]}{\Omega }\right){\left({\lambda }_{Q}Q+{\lambda }_R{\mathrm{I}}_p\right)}^{-1}\mid -\stackrel{[\mathrm{k}]\mathsf{T}}{q}{\left({\lambda }_{Q}Q+{\lambda }_R{\mathrm{I}}_p+\stackrel{[\mathrm{k}]}{\Omega }\right)}^{-1}\stackrel{[\mathrm{k}]}{q}\right\};\hfill \\ \hfill & \\ 8.k\leftarrow k+1;\hfill \\ \hfill & \mathbf{while}\left\{\Vert \stackrel{[\mathrm{k}]}{\lambda }-\stackrel{[\mathrm{k}-1]}{\lambda }\Vert ∕\Vert \stackrel{[\mathrm{k}-1]}{\lambda }\Vert >\delta \right\}\hfill \end{array} \end{gathered}$$