  In many applications it is useful to replace the Moore-Penrose pseudoinverse
(MPP) by a different generalized inverse with more favorable properties. We may
want, for example, to have many zero entries, but without giving up too much of
the stability of the MPP. One way to quantify stability is by how much the
Frobenius norm of a generalized inverse exceeds that of the MPP. In this paper
we derive finite-size concentration bounds for the Frobenius norm of
$\ell^p$-minimal general inverses of iid Gaussian matrices, with $1 \leq p \leq
2$. For $p = 1$ we prove exponential concentration of the Frobenius norm of the
sparse pseudoinverse; for $p = 2$, we get a similar concentration bound for the
MPP. Our proof is based on the convex Gaussian min-max theorem, but unlike
previous applications which give asymptotic results, we derive finite-size
bounds.
