Optimal Estimation of Shared Singular Subspaces across Multiple Noisy Matrices

2026-07-14 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, extended

Summary

A new study addresses the optimal estimation of shared singular subspaces from multiple noisy matrices, a critical challenge in data integration and multi-view analysis. The research establishes that the common Stack-SVD approach, which concatenates individual matrices, achieves minimax rate-optimality when true singular subspaces are identical. The analysis identifies phase transition phenomena related to signal-to-noise ratio and characterizes Stack-SVD's performance in scenarios with partially shared subspaces. To overcome Stack-SVD's limitations in these complex cases, the authors propose novel minimax rate-optimal estimators and an efficient algorithm for identifying shared and unshared singular vectors. Extensive simulations, including settings with n=10,20 and various (p1,p2) dimensions, and real-world applications to single-cell RNA-seq datasets (10x Chromium v3/v2, Drop-seq) demonstrate the proposed methods' advantages, showing up to 160% higher Calinski-Harabasz Index scores.

Key takeaway

For Data Scientists integrating noisy multi-view datasets, recognize Stack-SVD's optimal performance for fully shared singular subspaces. Its efficacy diminishes with partial sharing, however. You should consider implementing the proposed shared-signal tracing algorithm, particularly when individual SVDs struggle with non-identifiable vectors. This method significantly enhances the accuracy of shared latent structure estimation, as demonstrated by superior clustering metrics in real-world single-cell data integration tasks.

Key insights

Stack-SVD is minimax optimal for fully shared singular subspaces, but new algorithms are essential for partial sharing.

Principles

• Stack-SVD is minimax rate-optimal for identical singular subspaces.
• Minimax optimality depends on SNR and matrix dimensions.
• Sufficient eigen-gaps are crucial for distinguishing vector types.

Method

An algorithm identifies shared and unshared singular vectors by comparing distances between singular subspaces of stacked and individual noisy matrices.

In practice

• Integrate multi-modal single-cell data for joint dimensionality reduction.
• Identify latent gene structures across different sequencing technologies.

Topics

• Singular Value Decomposition
• Low-Rank Matrix Denoising
• Multi-view Data Integration
• Subspace Estimation
• Minimax Rates
• Single-cell RNA-seq

Best for: AI Scientist, Research Scientist, Data Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.