  We prove convergence guarantees for generalized low-rank matrix sensing --
i.e., where matrix sensing where the observations may be passed through some
nonlinear link function. We focus on local convergence of the optimal
estimator, ignoring questions of optimization. In particular, assuming the
minimizer of the empirical loss $\theta^0$ is in a constant size ball around
the true parameters $\theta^*$, we prove that
$d(\theta^0,\theta^*)=\tilde{O}(\sqrt{dk^2/n})$. Our analysis relies on tools
from Riemannian geometry to handle the rotational symmetry in the parameter
space.
