# Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence

arXiv: https://arxiv.org/abs/2105.13346 · HTML: https://mlsanity.com/paper/2105.13346/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2021.

## Built on by (most cited first, 12 of 12)

- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [Leave-one-out Singular Subspace Perturbation Analysis for Spectral Clustering](https://mlsanity.com/paper/2205.14855/) (2022)
- [Spectral Entry-wise Matrix Estimation for Low-Rank Reinforcement Learning](https://mlsanity.com/paper/2310.06793/) (2023)
- [Estimating Higher-Order Mixed Memberships via the ℓ2,∞ Tensor Perturbation Bound](https://mlsanity.com/paper/2212.08642/) (2022)
- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
- [Enhancing Accuracy in Deep Learning Using Random Matrix Theory](https://mlsanity.com/paper/2310.03165/) (2023)
- [Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methods](https://mlsanity.com/paper/2405.13912/) (2024)
- [Entrywise Recovery Guarantees for Sparse PCA via Sparsistent Algorithms](https://mlsanity.com/paper/2202.04061/) (2022)
- [On Minimum Trace Factor Analysis -- An Old Song Sung to a New Tune](https://mlsanity.com/paper/2402.02459/) (2024)
- [Generalized Grade-of-Membership Estimation for High-dimensional Locally Dependent Data](https://mlsanity.com/paper/2412.19796/) (2024)
- [Adjacency Spectral Embeddings of Correlation Networks](https://mlsanity.com/paper/2602.21055/) (2026)
- [Heteroskedastic Canonical Polyadic Tensor Decomposition](https://mlsanity.com/paper/2610.00498/) (2026)

## Abstract (arXiv)

We propose an estimator for the singular vectors of high-dimensional low-rank matrices corrupted by additive subgaussian noise, where the noise matrix is allowed to have dependence within rows and heteroskedasticity between them. We prove finite-sample ℓ2,∞ bounds and a Berry-Esseen theorem for the individual entries of the estimator, and we apply these results to high-dimensional mixture models. Our Berry-Esseen theorem clearly shows the geometric relationship between the signal matrix, the covariance structure of the noise, and the distribution of the errors in the singular vector estimation task. These results are illustrated in numerical simulations. Unlike previous results of this type, which rely on assumptions of gaussianity or independence between the entries of the additive noise, handling the dependence between entries in the proofs of these results requires careful leave-one-out analysis and conditioning arguments. Our results depend only on the signal-to-noise ratio, the sample size, and the spectral properties of the signal matrix.