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

2024-10-24

arXiv.org

AI Summary

In multi-view subspace learning, theoretical guarantees for distinguishing shared and individual signal subspaces from high-dimensional noisy data remain lacking. This paper establishes, for the first time, necessary and sufficient conditions for subspace separability and develops a rigorous theoretical framework based on spectral perturbation analysis of projection matrix products. Integrating rotational bootstrap with random matrix theory, we propose a parameter-free, interpretable method that automatically partitions subspaces into three categories—shared, individual, and noise—without manual tuning. Leveraging principal angle analysis and diagnostic visualization, our approach enhances estimation robustness. Extensive simulations demonstrate substantial improvements in estimation accuracy for both joint and individual subspaces over state-of-the-art methods. On real-world multi-omics colorectal cancer and murine nutritional genomics datasets, downstream classification and prediction performance is significantly enhanced.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Intelligent Robots: State Estimation
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Identify joint and individual signal subspaces in multi-view data
• Quantify conditions for reliable subspace separation
• Develop scalable algorithm for subspace estimation

Innovation

• Product of projections for subspace learning
• Rotational bootstrap for spectrum partitioning
• Diagnostic plots for estimation visualization