# What type of matrices is approximate inverse preconditioner $||I - AM||_F$ well suited for?

Take sparse approximate inverse preconditioner $M \approx A^{-1}$ given by solution of

$$\underset{M \in S}{\mbox{min}} \; ||I - AM||_F,$$

where $S$ is a set of sparse matrices and $||.||_F$ is the Frobenius norm (see [1] for details).

I am wondering which specific characteristics of the matrix would make this preconditioner very efficient? For instance, it is not difficult to see why Jacobi scaling is efficient for diagonally dominant matrices, SOR for tridiagonal matrices, multigrid for discretizated elliptic operators, etc. But I am lacking an insight into how the preconditioner above could implicitly or explicitly exploit any particular (perhaps structure-related?) characteristics of the matrix.