# Nearly Optimal Stochastic Approximation for Online Principal Subspace Estimation

arXiv: https://arxiv.org/abs/1711.06644 · HTML: https://mlsanity.com/paper/1711.06644/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2017.

## Built on by (most cited first, 2 of 2)

- [On the Optimality of the Oja's Algorithm for Online PCA](https://mlsanity.com/paper/2104.00512/) (2021)
- [Global Convergence of Adaptive Sensing for Principal Eigenvector Estimation](https://mlsanity.com/paper/2505.10882/) (2025)

## Abstract (arXiv)

Principal component analysis (PCA) has been widely used in analyzing high-dimensional data. It converts a set of observed data points of possibly correlated variables into a set of linearly uncorrelated variables via an orthogonal transformation. To handle streaming data and reduce the complexities of PCA, (subspace) online PCA iterations were proposed to iteratively update the orthogonal transformation by taking one observed data point at a time. Existing works on the convergence of (subspace) online PCA iterations mostly focus on the case where sample are almost surely uniformly bounded. In this paper, we analyze the convergence of a subspace online PCA iteration under more practical assumption and obtain a nearly optimal finite-sample error bound. Our convergence rate almost matches the minimax information lower bound. We prove that the convergence is nearly global in the sense that the subspace online PCA iteration is convergent with high probability for random initial guesses. This work also leads to a simpler proof of the recent work on analyzing online PCA for the first principal component only.