In the usual VIF calculation for a linear regression, each independent/explanatory variable $X_j$ is treated as the dependent variable in an ordinary least squares regression. i.e.
$$ X_j = \beta_0 + \sum_{i=1, i \neq j}^n \beta_i X_i $$
The $R^2$ values are stored for each of the $n$ regressions and VIF is determined by
$$ VIF_j = \frac{1}{1-R^2_j} $$
for a particular explanatory variable.
Suppose my generalize additive model takes the form, $$ Y=\beta_0+ \sum_{i=1}^n \beta_iX_i + \sum_{j=1}^m s_j(X_i) . $$
Is there an equivalent VIF calculation for this type of model? Is there a way I can control for the smooth terms $s_j$ to test for multicollinearity?