Alternating minimization for generalized rank-1 matrix sensing: sharp predictions from a random initialization

Abstract 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 non-convex optimization problem starting from a random initialization. We show sharp convergence guarantees for a sample-split version of the algorithm by deriving a deterministic one-step 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 one-step 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 behaviour. On a technical level, our results are enabled by showing that the empirical error recursion can be predicted by our deterministic one-step updates 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 complex iterative algorithms from a random initialization in other high-dimensional optimization problems with random data.