用几何方法解决噪声下低秩张量恢复难题
Guaranteed Noisy CP Tensor Recovery via Riemannian Optimization on the Segre Manifold
- 在塞格雷流形上用黎曼优化求解张量恢复
- 梯度下降法线性收敛，高斯牛顿法前阶段二次收敛
- 适合做张量数据分析的科研人员参考
从噪声线性测量中恢复低CP秩张量是高维数据分析的核心挑战，应用涵盖张量主成分分析、张量回归等。本文利用秩1张量的内在几何结构，将恢复问题转化为塞格雷流形（Segre manifold）上的优化问题，该流形是秩1张量构成的光滑黎曼流形。由此导出两种高效算法：黎曼梯度下降（RGD）与黎曼高斯牛顿法（RGN），二者均保证每步迭代保持可行性。在温和噪声假设下，证明RGD具有局部线性收敛速率，而RGN在初始阶段呈现局部二次收敛，随后过渡至线性收敛，直至逼近统计噪声水平。大量合成实验验证了这些收敛性理论，并展示了方法的实际有效性。
原文摘要 · Abstract (English)
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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