When comparing GMM models with different number of components (i.e number of Gaussians) one penalizes the likelihood for the total number of free parameters in the mixture model. If the data is in $D$ dimension then the number of free parameters for $J$ components is given as:
$J-1$: for $J$ weights which sum to one
$D$: for each mean
$D(D+1)/2$: for each covariance matrix
My question is this: assuming that I am working in a limited space in $D$ and I do not need infinite precision, if I discretize my space into grid points instead of referring to each mean location with its coordinate vector of $D$ elements can't I just use one parameter as an index to define the location? It would work only for the mean vector but nevertheless it would be a reduction of complexity.