  We consider the problem of estimating the factors of a rank-$1$ matrix with
i.i.d. Gaussian, rank-$1$ measurements that are nonlinearly transformed and
corrupted by noise. Considering two prototypical choices for the nonlinearity,
we study the convergence properties of a natural alternating update rule for
this nonconvex optimization problem starting from a random initialization. We
show sharp convergence guarantees for a sample-split version of the algorithm
by deriving a deterministic recursion that is accurate even in high-dimensional
problems. Notably, while the infinite-sample population update is uninformative
and suggests exact recovery in a single step, the algorithm -- and our
deterministic prediction -- converges geometrically fast from a random
initialization. Our sharp, non-asymptotic analysis also exposes several other
fine-grained properties of this problem, including how the nonlinearity and
noise level affect convergence behavior.
  On a technical level, our results are enabled by showing that the empirical
error recursion can be predicted by our deterministic sequence within
fluctuations of the order $n^{-1/2}$ when each iteration is run with $n$
observations. Our technique leverages leave-one-out tools originating in the
literature on high-dimensional $M$-estimation and provides an avenue for
sharply analyzing higher-order iterative algorithms from a random
initialization in other high-dimensional optimization problems with random
data.
