基于Segre流形上黎曼优化的有保证噪声CP张量恢复
Guaranteed Noisy CP Tensor Recovery via Riemannian Optimization on the Segre Manifold
AI总结：
针对高维数据分析中低CP秩张量的含噪恢复难题，提出基于Segre流形黎曼优化的RGD与RGN算法，证明其收敛性并通过实验验证了有效性。
摘要：
从含噪线性测量中恢复低CP秩张量是高维数据分析的核心挑战，应用涵盖张量PCA、张量回归等诸多领域。我们利用秩一张量的内蕴几何结构，将恢复任务建模为Segre流形（秩一张量构成的光滑黎曼流形）上的优化问题。该几何视角催生了两种高效算法：黎曼梯度下降（RGD）和黎曼高斯-牛顿（RGN），二者在每次迭代中均保持可行性。在温和噪声假设下，我们证明RGD以局部线性速率收敛，而RGN在迭代初期呈现局部二次收敛阶段，当迭代点接近统计噪声基底时转为线性收敛速率。大量合成实验验证了这些收敛保证，也证明了我们方法的实际有效性。
英文摘要：
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.
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