标题： 基于黎曼优化的塞格雷流形上保证的噪声CP张量恢复 显示英文标题

标题： Guaranteed Noisy CP Tensor Recovery via Riemannian Optimization on the Segre Manifold

摘要： 从带有噪声的线性测量中恢复低-CP-秩张量是高维数据分析中的一个核心挑战，其应用范围涵盖张量PCA、张量回归等。我们通过将恢复任务转化为Segre流形上的优化问题，利用了秩一张量的内在几何结构，Segre流形是秩一张量的光滑黎曼流形。这种几何观点产生了两种强大的算法：黎曼梯度下降（RGD）和黎曼高斯-牛顿（RGN），每种算法在每次迭代中都保持可行性。在轻微噪声假设下，我们证明RGD以局部线性速率收敛，而RGN在初始阶段表现出局部二次收敛，随着迭代接近统计噪声水平，收敛速率过渡到线性。广泛的合成实验验证了这些收敛保证，并展示了我们方法的实际有效性。 显示英文摘要

摘要： Recovering a low-CP-rank tensor from noisy linear measurements is a central challenge in high-dimensional data analysis, with applications spanning tensor PCA, tensor regression, and beyond. We exploit the intrinsic geometry of rank-one tensors by casting the recovery task as an optimization problem over the Segre manifold, the smooth Riemannian manifold of rank-one tensors. This geometric viewpoint yields two powerful algorithms: Riemannian Gradient Descent (RGD) and Riemannian Gauss-Newton (RGN), each of which preserves feasibility at every iteration. Under mild noise assumptions, we prove that RGD converges at a local linear rate, while RGN exhibits an initial local quadratic convergence phase that transitions to a linear rate as the iterates approach the statistical noise floor. Extensive synthetic experiments validate these convergence guarantees and demonstrate the practical effectiveness of our methods.