On the application of the symmetric graphical lasso for paired data

Graphical lasso methods are not invariant to scalar multiplication of the variables. On the other hand, Gaussian graphical models are invariant to scalar multiplication, and thus it is common practice to apply graphical lasso after the observed variables are standardized to unit sample variances. We consider the symmetric graphical lasso method for learning Gaussian graphical models for paired data and show that this family of models is not invari- ant to scalar multiplication of the variables, but that in the special case where homologous variables have equal variances it still makes sense to standardise the variables. We then carry out an empirical analysis to assess the impact of standardization on the symmetric graphical lasso method.