# A spectral method for multi-view subspace learning using the product of projections

arXiv: https://arxiv.org/abs/2410.19125 · HTML: https://mlsanity.com/paper/2410.19125/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 30 references that are in the map.
- First posted on arXiv 24 Oct 2024.

## Builds on (most cited first, 11 of 11)

- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [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)
- [Joint association and classification analysis of multi-view data](https://mlsanity.com/paper/1811.08511/) (2018)
- [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)
- [Integrative Factorization of Bidimensionally Linked Matrices](https://mlsanity.com/paper/1906.03722/) (2019)
- [False Discovery and Its Control in Low Rank Estimation](https://mlsanity.com/paper/1810.08595/) (2018)
- [Exponential canonical correlation analysis with orthogonal variation](https://mlsanity.com/paper/2208.00048/) (2022)
- [Sparse and Integrative Principal Component Analysis for Multiview Data](https://mlsanity.com/paper/2301.06718/) (2023)

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

- [Optimal Estimation of Shared Singular Subspaces across Multiple Noisy Matrices](https://mlsanity.com/paper/2411.17054/) (2024)
- [Stacked SVD or SVD stacked? A Random Matrix Theory perspective on data integration](https://mlsanity.com/paper/2507.22170/) (2025)
- [Multi-layer Cross-attention is Provably Optimal for Multi-modal In-context Learning](https://mlsanity.com/paper/2602.04872/) (2026)
- [PLATONT: Learning a Platonic Representation for Unified Network Tomography](https://mlsanity.com/paper/2511.15251/) (2025)
- [FiGuRO: Intrinsic Dimension Estimation for Multi-Modal Data](https://mlsanity.com/paper/2608.10857/) (2026)

## Abstract (arXiv)

Multi-view data provides complementary information on the same set of observations, with multi-omics and multimodal sensor data being common examples. Analyzing such data typically requires distinguishing between shared (joint) and unique (individual) signal subspaces from noisy, high-dimensional measurements. Despite many proposed methods, the conditions for reliably identifying joint and individual subspaces remain unclear. We rigorously quantify these conditions, which depend on the ratio of the signal rank to the ambient dimension, principal angles between true subspaces, and noise levels. Our approach characterizes how spectrum perturbations of the product of projection matrices, derived from each view's estimated subspaces, affect subspace separation. Using these insights, we provide an easy-to-use and scalable estimation algorithm. In particular, we employ rotational bootstrap and random matrix theory to partition the observed spectrum into joint, individual, and noise subspaces. Diagnostic plots visualize this partitioning, providing practical and interpretable insights into the estimation performance. In simulations, our method estimates joint and individual subspaces more accurately than existing approaches. Applications to multi-omics data from colorectal cancer patients and nutrigenomic study of mice demonstrate improved performance in downstream predictive tasks.