the purpose is to never store a full X with n_sources, n_times but just the non zeros. In other words we should never have to do : np.zeros((n_sources, n_times))

use init with (X, active set) is a simple way to have a sparse row matrix.

but you already transfer X restricted to the active set (both _solver functions do this restriction) and G[:, active_set] to the _solver functions. So active_set (which is defined on the full leadfield) has no real meaning in the _solver functions as it points to the full leadfield.

In the first iteration, X will be a dense matrix, but then as soon as we use init, X will be nonzero in rows corresponding to the old active set

In the init case, the matrix I pass, is still the old one, which is a matrix n_new_as x n_times, with non_zeros in n_old_as rows and zeros otherwise