# ROP: Matrix recovery via rank-one projections

arXiv: https://arxiv.org/abs/1310.5791 · HTML: https://mlsanity.com/paper/1310.5791/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2013.

## Built on by (most cited first, 18 of 18)

- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Flat minima generalize for low-rank matrix recovery](https://mlsanity.com/paper/2203.03756/) (2022)
- [A General Algorithm for Solving Rank-one Matrix Sensing](https://mlsanity.com/paper/2303.12298/) (2023)
- [Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence](https://mlsanity.com/paper/2104.12031/) (2021)
- [Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization](https://mlsanity.com/paper/2108.01772/) (2021)
- [Computationally Efficient and Statistically Optimal Robust High-Dimensional Linear Regression](https://mlsanity.com/paper/2305.06199/) (2023)
- [Perceptual adjustment queries and an inverted measurement paradigm for low-rank metric learning](https://mlsanity.com/paper/2309.04626/) (2023)
- [Theoretical Guarantees for Low-Rank Compression of Deep Neural Networks](https://mlsanity.com/paper/2502.02766/) (2025)
- [Provably Learning Multi-Head Attention with Queries](https://mlsanity.com/paper/2608.03294/) (2026)
- [ROP inception: signal estimation with quadratic random sketching](https://mlsanity.com/paper/2205.08225/) (2022)
- [Learning the Structure of Large Networked Systems Obeying Conservation Laws](https://mlsanity.com/paper/2206.07083/) (2022)
- [Compressive Recovery of Sparse Precision Matrices](https://mlsanity.com/paper/2311.04673/) (2023)
- [Provably Learning Attention with Queries](https://mlsanity.com/paper/2601.16873/) (2026)
- [Matrix Completion via Residual Spectral Matching](https://mlsanity.com/paper/2412.10005/) (2024)
- [A Smoothing Newton Method for Rank-one Matrix Recovery](https://mlsanity.com/paper/2507.23017/) (2025)
- [Robust Sparse Signal Recovery with Outliers: A Hard Thresholding Pursuit Approach Based on LAD](https://mlsanity.com/paper/2601.06558/) (2026)
- [SCaLE: Switching Cost aware Learning and Exploration](https://mlsanity.com/paper/2601.09042/) (2026)
- [Random features for Grassmannian kernel approximation with bounded rank-one projections](https://mlsanity.com/paper/2608.04227/) (2026)

## Abstract (arXiv)

Estimation of low-rank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rank-one projection model for low-rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case. The procedure is adaptive to the rank and robust against small perturbations. Both upper and lower bounds for the estimation accuracy under the Frobenius norm loss are obtained. The proposed estimator is shown to be rate-optimal under certain conditions. The estimator is easy to implement via convex programming and performs well numerically. The techniques and main results developed in the paper also have implications to other related statistical problems. An application to estimation of spiked covariance matrices from one-dimensional random projections is considered. The results demonstrate that it is still possible to accurately estimate the covariance matrix of a high-dimensional distribution based only on one-dimensional projections.