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

2025-10-01

AI Summary

This paper addresses low-rank tensor recovery from noisy linear measurements, aiming to improve accuracy and efficiency in high-dimensional data analysis tasks such as tensor PCA and tensor regression. We propose a Riemannian optimization framework built on the Segre manifold—the set of rank-one tensors—which naturally encodes the low-canonical-polyadic (CP) rank constraint via the manifold’s geometric structure, ensuring all iterates remain feasible. To our knowledge, this is the first work to design and analyze Riemannian gradient descent (RGD) and Riemannian Gauss–Newton (RGN) algorithms for this problem with rigorous convergence guarantees: RGD achieves local linear convergence, while RGN attains local quadratic convergence when initialized sufficiently close to the ground truth. Experiments demonstrate that both methods are stable and efficient across diverse noise levels and achieve statistically optimal estimation rates.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Recovering low-CP-rank tensors from noisy linear measurements
• Solving tensor recovery via Riemannian optimization on Segre manifold
• Developing algorithms with guaranteed convergence under noise conditions

Innovation

• Optimization on Segre manifold for tensor recovery
• Riemannian Gradient Descent with linear convergence rate
• Riemannian Gauss-Newton with quadratic-linear convergence transition