Algorithm 1 GLEN

$$\begin{array}{cc}\hfill & \mathrm{Input}:\text{noisy signals}\mathbf{X}=[{\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_M],\text{exponential}\hfill \\ \hfill & \text{family distribution}𝒫,\text{trade-off parameters}{\alpha }_1,{\alpha }_2,\text{and}\gamma ,\hfill \\ \hfill & \text{stepsize}\rho \text{and}\lambda \hfill \\ \hfill & \mathrm{Output}:\text{graph Laplacian}\mathbf{L},\text{smooth signal representation}\hfill \\ \hfill & \mathbf{Y}=[{\mathbf{y}}_1,{\mathbf{y}}_2,\dots ,{\mathbf{y}}_M]\hfill \\ \hfill & \text{Initialize}\mathbf{Y}\text{and}\mathbf{L},\text{or}\mathbf{Y},\mathbf{L},\text{and}\mu \text{if using the offset}\hfill \\ \hfill & \mathbf{repeat}\hfill \\ \hfill & \text{Solve}\mathbf{L}\text{in}(17)\text{using gradient descent}\hfill \\ \hfill & \mathbf{for}j=1,2,\dots ,M\mathbf{do}\hfill \\ \hfill & \text{Calculate}{\nabla }_{{\mathbf{y}}_j}\text{and}{\nabla }_{{\mathbf{y}}_j}^2\text{as in}(20)\text{and}(21),(\text{or}(39)\hfill \\ \hfill & \text{and}(40)\text{when using time-vertex signals})\hfill \\ \hfill & \text{Solve}\Delta {\mathbf{y}}_j\text{from}(22)\text{and update}{\mathbf{y}}_j\text{as in}(23)\hfill \\ \hfill & \mathbf{end}\mathbf{for}\hfill \\ \hfill & \text{Update}\mu \text{as in}(33)\text{if using the offset}\hfill \\ \hfill & \text{Calculate the loss using}(16)\hfill \\ \hfill & \mathbf{until}\text{convergence}.\hfill \end{array}$$