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

arXiv: https://arxiv.org/abs/2510.00569 · HTML: https://mlsanity.com/paper/2510.00569/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 33 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## Builds on (most cited first, 13 of 13)

- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [An Optimal Statistical and Computational Framework for Generalized Tensor Estimation](https://mlsanity.com/paper/2002.11255/) (2020)
- [Tensor Methods and Recommender Systems](https://mlsanity.com/paper/1603.06038/) (2016)
- [Tensor-on-tensor regression](https://mlsanity.com/paper/1701.01037/) (2017)
- [Guaranteed Non-Orthogonal Tensor Decomposition via Alternating Rank-1 Updates](https://mlsanity.com/paper/1402.5180/) (2014)
- [Provable Sparse Tensor Decomposition](https://mlsanity.com/paper/1502.01425/) (2015)
- [Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay](https://mlsanity.com/paper/2206.08756/) (2022)
- [ISLET: Fast and Optimal Low-rank Tensor Regression via Importance Sketching](https://mlsanity.com/paper/1911.03804/) (2019)
- [Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality](https://mlsanity.com/paper/2006.08580/) (2020)
- [Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence](https://mlsanity.com/paper/2104.12031/) (2021)
- [Tensor Principal Component Analysis in High Dimensional CP Models](https://mlsanity.com/paper/2108.04428/) (2021)
- [Multivariate Temporal Point Process Regression](https://mlsanity.com/paper/2001.00719/) (2020)
- [Tensor completion using geodesics on Segre manifolds](https://mlsanity.com/paper/2108.00735/) (2021)

## Abstract (arXiv)

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.