# Optimal Estimation of Shared Singular Subspaces across Multiple Noisy Matrices

arXiv: https://arxiv.org/abs/2411.17054 · HTML: https://mlsanity.com/paper/2411.17054/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 48 references that are in the map.
- First posted on arXiv 26 Nov 2024.

## Builds on (most cited first, 17 of 17)

- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Denoising Implicit Feedback for Recommendation](https://mlsanity.com/paper/2006.04153/) (2020)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [Angle-Based Joint and Individual Variation Explained](https://mlsanity.com/paper/1704.02060/) (2017)
- [Structural Learning and Integrative Decomposition of Multi-View Data](https://mlsanity.com/paper/1707.06573/) (2017)
- [sJIVE: Supervised Joint and Individual Variation Explained](https://mlsanity.com/paper/2102.13278/) (2021)
- [Optimal Permutation Recovery in Permuted Monotone Matrix Model](https://mlsanity.com/paper/1911.10604/) (2019)
- [Hierarchical nuclear norm penalization for multi-view data](https://mlsanity.com/paper/2206.12891/) (2022)
- [Data Integration Via Analysis of Subspaces (DIVAS)](https://mlsanity.com/paper/2212.00703/) (2022)
- [Optimal Structured Principal Subspace Estimation: Metric Entropy and Minimax Rates](https://mlsanity.com/paper/2002.07624/) (2020)
- [A spectral method for multi-view subspace learning using the product of projections](https://mlsanity.com/paper/2410.19125/) (2024)
- [Integrative decomposition of multi-source data by identifying partially-joint score subspaces](https://mlsanity.com/paper/2203.14041/) (2022)
- [RaJIVE: Robust Angle Based JIVE for Integrating Noisy Multi-Source Data](https://mlsanity.com/paper/2101.09110/) (2021)
- [Two sample test for covariance matrices in ultra-high dimension](https://mlsanity.com/paper/2312.10796/) (2023)
- [D-CDLF: Decomposition of Common and Distinctive Latent Factors for Multi-view High-dimensional Data](https://mlsanity.com/paper/2407.00730/) (2024)

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

- [Stacked SVD or SVD stacked? A Random Matrix Theory perspective on data integration](https://mlsanity.com/paper/2507.22170/) (2025)
- [Inference on covariance structure in high-dimensional multi-view data](https://mlsanity.com/paper/2509.02772/) (2025)
- [GlowQ: Group-Shared LOw-Rank Approximation for Quantized LLMs](https://mlsanity.com/paper/2603.25385/) (2026)
- [Enhancing Spectral Embedding through Robust and Flexible Knowledge Transfer in Electronic Health Records](https://mlsanity.com/paper/2606.11570/) (2026)
- [Unveiling Invariant and Transferable Latent Factors Across Heterogeneous Environments via ATLAS](https://mlsanity.com/paper/2607.18209/) (2026)
- [DAIF: A Data-Driven Intermediate Fusion Framework for Multimodal Supervised Learning via Approximate Message Passing](https://mlsanity.com/paper/2608.02769/) (2026)

## Abstract (arXiv)

Estimating singular subspaces from noisy matrices is a fundamental problem with wide-ranging applications across various fields. Driven by the challenges of data integration and multi-view analysis, this study focuses on estimating shared singular subspaces across multiple matrices within a low-rank matrix denoising framework. A common approach for this task is to perform singular value decomposition on the stacked matrix (Stack-SVD), which concatenates all the matrices. We establish that Stack-SVD achieves minimax rate-optimality when the true singular subspaces of the noisy matrices are identical, whereas a popular alternative approach based on SVD of concatenated singular vector matrices (Average-SVD) can be sub-optimal. We then tackle the more complex scenario where the true singular subspaces are only partially shared across matrices. For various cases of partial sharing, we rigorously characterize the conditions under which Stack-SVD remains effective, achieves minimax optimality, or fails to deliver consistent estimates, offering theoretical insights into its practical applicability. To address the limitations of Stack-SVD in scenarios with partial sharing, we propose novel estimators and an efficient algorithm designed to identify both shared and unshared singular vectors. We further prove that these methods attain minimax rate-optimality under partial sharing. Extensive simulations and real-world data applications demonstrate the advantages of our proposed approach.