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.

翻译：我们证明了广义低秩矩阵感知的收敛性保证——即观测数据可能通过某些非线性链接函数传递的矩阵感知问题。我们聚焦于最优估计量的局部收敛性，暂不讨论优化过程的具体问题。特别地，在假设经验损失最小化解 $\theta^0$ 位于真实参数 $\theta^*$ 的常数半径邻域内时，我们证明了 $d(\theta^0,\theta^*)=\tilde{O}(\sqrt{dk^2/n})$。我们的分析利用黎曼几何工具来处理参数空间中的旋转对称性。