# Stacked SVD or SVD stacked? A Random Matrix Theory perspective on data integration

arXiv: https://arxiv.org/abs/2507.22170 · HTML: https://mlsanity.com/paper/2507.22170/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 57 references that are in the map.
- First posted on arXiv 29 Jul 2025.

## Builds on (most cited first, 12 of 12)

- [Optimality of Spectral Clustering in the Gaussian Mixture Model](https://mlsanity.com/paper/1911.00538/) (2019)
- [Inference for multiple heterogeneous networks with a common invariant subspace](https://mlsanity.com/paper/1906.10026/) (2019)
- [Personalized PCA: Decoupling Shared and Unique Features](https://mlsanity.com/paper/2207.08041/) (2022)
- [Estimating shared subspace with AJIVE: the power and limitation of multiple data matrices](https://mlsanity.com/paper/2501.09336/) (2025)
- [A Spectral Method for Assessing and Combining Multiple Data Visualizations](https://mlsanity.com/paper/2210.13711/) (2022)
- [Is your data alignable? Principled and interpretable alignability testing and integration of single-cell data](https://mlsanity.com/paper/2308.01839/) (2023)
- [Optimal Estimation of Shared Singular Subspaces across Multiple Noisy Matrices](https://mlsanity.com/paper/2411.17054/) (2024)
- [Kernel spectral joint embeddings for high-dimensional noisy datasets using duo-landmark integral operators](https://mlsanity.com/paper/2405.12317/) (2024)
- [A spectral method for multi-view subspace learning using the product of projections](https://mlsanity.com/paper/2410.19125/) (2024)
- [Bayesian Combinatorial Multi-Study Factor Analysis](https://mlsanity.com/paper/2007.12616/) (2020)
- [Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets](https://mlsanity.com/paper/2407.01718/) (2024)
- [A Generalized Mean Approach for Distributed-PCA](https://mlsanity.com/paper/2410.00397/) (2024)

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

- [Information bottleneck for learning the phase space of dynamics from high-dimensional experimental data](https://mlsanity.com/paper/2604.24662/) (2026)
- [Transfer Learning in High-Dimensional Clustering: Minimax Thresholds and Applications in Single-Cell Data](https://mlsanity.com/paper/2607.25031/) (2026)
- [Multi-context principal component analysis](https://mlsanity.com/paper/2601.15239/) (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)
- [Bias-Corrected Subspace Intersection: Minimax-Optimal Shared Subspace Estimation in Multi-View Data](https://mlsanity.com/paper/2609.05617/) (2026)

## Abstract (arXiv)

Modern data analysis increasingly requires identifying shared latent structure across multiple high-dimensional datasets. A commonly used model assumes that the data matrices are noisy observations of low-rank matrices with a shared singular subspace. In this case, two primary methods have emerged for estimating this shared structure, which vary in how they integrate information across datasets. The first approach, termed Stack-SVD, concatenates all the datasets, and then performs a singular value decomposition (SVD). The second approach, termed SVD-Stack, first performs an SVD separately for each dataset, then aggregates the top singular vectors across these datasets, and finally computes a consensus amongst them. While these methods are widely used, they have not been rigorously studied in the proportional asymptotic regime, which is of great practical relevance in today's world of increasing data size and dimensionality. Consequently, it remains unclear when one method should be preferred over another. In this work, we derive exact expressions for the asymptotic performance and phase transitions of these two methods and develop optimal weighting schemes to further improve both methods. Our analysis reveals that while neither method uniformly dominates the other in the unweighted case, optimally weighted Stack-SVD dominates optimally weighted SVD-Stack when the low rank signal is fully shared across the datasets. We then analyze multiple, partially shared components per dataset and demonstrate that SVD-Stack can yield improved performance without requiring estimation of subspace alignment. Finally, we provide practical algorithms for estimating optimal weights from data, offering theoretical guidance for method selection in practical data integration problems. Extensive numerical simulations and semi-synthetic experiments on genomic data corroborate our theoretical findings.