← 研踪 PaperTrailarXiv:2510.00569 · v2
Guaranteed Noisy CP Tensor Recovery via Riemannian Optimization on the Segre Manifold
首次提交 2025-10-01 14:44
版本更新 2026-11-08 07:24
雷达观测 2026-11-10 12:16
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, scalar-on-tensor regression, tensor-on-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 phase followed by linear refinement to the statistical error. Extensive synthetic experiments validate these convergence guarantees and demonstrate the practical effectiveness of our methods.
作者备注：Accepted at NeurIPS 2026 (camera-ready version). 49 pages, 5 figures, 10 tables. Code: https://github.com/kxu-stat/SegreCP
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v2 · 2026-11-08 07:24 · 最新观测版本
Guaranteed Noisy CP Tensor Recovery via Riemannian Optimization on the Segre Manifold
Recovering a low-CP-rank tensor from noisy linear measurements is a central challenge in high-dimensional data analysis, with applications spanning tensor PCA, scalar-on-tensor regression, tensor-on-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 phase followed by linear refinement to the statistical error. Extensive synthetic experiments validate these convergence guarantees and demonstrate the practical effectiveness of our methods.
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