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

TL;DR

This paper introduces a spectral method for multi-view subspace learning that accurately separates joint and individual signal subspaces in noisy, high-dimensional multi-view data, with applications in biomedical research.

Contribution

The paper provides a rigorous analysis of conditions for subspace identification and introduces a scalable spectral algorithm utilizing spectrum perturbation and bootstrap techniques.

Findings

More accurate estimation of joint and individual subspaces than existing methods

Effective application to multi-omics data improves downstream predictions

Diagnostic plots aid in interpreting subspace separation performance

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.