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

Summary Multi-view data provide complementary information on the same set of observations, with multi-omics and multimodal sensor data being common examples. Analysing 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, the 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, 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 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 a nutrigenomic study of mice demonstrate improved performance in downstream predictive tasks.