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 linear convergence guarantees for a sample-split version of the algorithm by deriving a deterministic recursion that is accurate even in high-dimensional problems. Our sharp, non-asymptotic analysis also exposes several other fine-grained properties of this problem, including how the nonlinearity, sample size, and noise level affect convergence behavior.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 and provides an avenue for sharply analyzing higher-order iterative algorithms from a random initialization in other optimization problems with random data.