A spectral method for multi-view subspace learning using the product of projections
This work addresses the challenge of analyzing multi-view data like multi-omics for researchers, providing a practical tool with interpretable diagnostics, though it is incremental as it builds on existing subspace learning methods.
The paper tackles the problem of reliably identifying joint and individual signal subspaces in multi-view data by quantifying conditions based on signal rank, principal angles, and noise levels, resulting in a scalable algorithm that outperforms existing methods in simulations and improves predictive tasks in multi-omics applications.
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.