# Subspace Estimation from Incomplete Observations: A High-Dimensional Analysis

arXiv: https://arxiv.org/abs/1805.06834 · HTML: https://mlsanity.com/paper/1805.06834/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2018.

## Built on by (most cited first, 6 of 6)

- [Phase diagram of Stochastic Gradient Descent in high-dimensional two-layer neural networks](https://mlsanity.com/paper/2202.00293/) (2022)
- [Learning Dynamics in Linear VAE: Posterior Collapse Threshold, Superfluous Latent Space Pitfalls, and Speedup with KL Annealing](https://mlsanity.com/paper/2310.15440/) (2023)
- [High-Dimensional Learning Dynamics of Quantized Models with Straight-Through Estimator](https://mlsanity.com/paper/2510.10693/) (2025)
- [Exploring the Precise Dynamics of Single-Layer GAN Models: Leveraging Multi-Feature Discriminators for High-Dimensional Subspace Learning](https://mlsanity.com/paper/2411.00498/) (2024)
- [Training Dynamics of Nonlinear Contrastive Learning Model in the High Dimensional Limit](https://mlsanity.com/paper/2406.06909/) (2024)
- [Effective Covariance Dynamics in Solvable High-Dimensional GANs](https://mlsanity.com/paper/2606.27246/) (2026)

## Abstract (arXiv)

We present a high-dimensional analysis of three popular algorithms, namely, Oja's method, GROUSE and PETRELS, for subspace estimation from streaming and highly incomplete observations. We show that, with proper time scaling, the time-varying principal angles between the true subspace and its estimates given by the algorithms converge weakly to deterministic processes when the ambient dimension n tends to infinity. Moreover, the limiting processes can be exactly characterized as the unique solutions of certain ordinary differential equations (ODEs). A finite sample bound is also given, showing that the rate of convergence towards such limits is O(1/√n). In addition to providing asymptotically exact predictions of the dynamic performance of the algorithms, our high-dimensional analysis yields several insights, including an asymptotic equivalence between Oja's method and GROUSE, and a precise scaling relationship linking the amount of missing data to the signal-to-noise ratio. By analyzing the solutions of the limiting ODEs, we also establish phase transition phenomena associated with the steady-state performance of these techniques.