# Optimal Structured Principal Subspace Estimation: Metric Entropy and Minimax Rates

arXiv: https://arxiv.org/abs/2002.07624 · HTML: https://mlsanity.com/paper/2002.07624/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Feb 2020.

## Built on by (most cited first, 5 of 5)

- [Leave-one-out Singular Subspace Perturbation Analysis for Spectral Clustering](https://mlsanity.com/paper/2205.14855/) (2022)
- [A Spectral Method for Assessing and Combining Multiple Data Visualizations](https://mlsanity.com/paper/2210.13711/) (2022)
- [Optimal Estimation of Shared Singular Subspaces across Multiple Noisy Matrices](https://mlsanity.com/paper/2411.17054/) (2024)
- [Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms](https://mlsanity.com/paper/2201.06438/) (2022)
- [Do algorithms and barriers for sparse principal component analysis extend to other structured settings?](https://mlsanity.com/paper/2307.13535/) (2023)

## Abstract (arXiv)

Driven by a wide range of applications, many principal subspace estimation problems have been studied individually under different structural constraints. This paper presents a unified framework for the statistical analysis of a general structured principal subspace estimation problem which includes as special cases non-negative PCA/SVD, sparse PCA/SVD, subspace constrained PCA/SVD, and spectral clustering. General minimax lower and upper bounds are established to characterize the interplay between the information-geometric complexity of the structural set for the principal subspaces, the signal-to-noise ratio (SNR), and the dimensionality. The results yield interesting phase transition phenomena concerning the rates of convergence as a function of the SNRs and the fundamental limit for consistent estimation. Applying the general results to the specific settings yields the minimax rates of convergence for those problems, including the previous unknown optimal rates for non-negative PCA/SVD, sparse SVD and subspace constrained PCA/SVD.