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.

翻译：我们考虑从经过非线性变换并受噪声污染的独立同分布高斯秩一测量中，估计秩一矩阵因子的问题。针对两种典型的非线性函数选择，我们研究了该非凸优化问题从随机初始化出发时，一种自然交替更新规则的收敛特性。通过推导即使在髙维问题中也精确的确定性递推关系，我们证明了该算法的样本分割版本具有严格的收敛保证。值得注意的是，尽管无限样本总体更新无法提供信息且暗示单步即可精确恢复，但该算法——以及我们的确定性预测——从随机初始化出发能以几何速率快速收敛。我们严格的非渐近分析还揭示了该问题的若干其他精细性质，包括非线性函数和噪声水平如何影响收敛行为。在技术层面，我们的结果通过证明：当每次迭代使用n个观测值时，经验误差递推可在n^{-1/2}量级的波动范围内由我们的确定性序列预测而实现。我们的技术利用了源于髙维M估计文献的留一法工具，为在其他具有随机数据的髙维优化问题中严格分析从随机初始化出发的高阶迭代算法提供了途径。