You almost never compute the inverse (pseudo or otherwise) of a sparse matrix because the inverse is generally dense.
On the other hand, you do compute the application of the inverse to a vector, and you often precompute factorizations that let you apply the inverse more quickly.
In the case of the ordinary inverse, you can apply a sparse inverse with A \ b and compute the factorization with lu(A) etcetera. In the case of the pseudoinverse, applying it computes the least-squares solution, and the corresponding factorization is usually qr.