{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T00:51:44Z","timestamp":1776473504714,"version":"3.51.2"},"reference-count":30,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T00:00:00Z","timestamp":1764892800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,2,13]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>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\u2019s 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.<\/jats:p>","DOI":"10.1093\/biomet\/asaf088","type":"journal-article","created":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T13:07:16Z","timestamp":1764940036000},"source":"Crossref","is-referenced-by-count":0,"title":["A spectral framework for multi-view subspace learning using the product of projections"],"prefix":"10.1093","volume":"113","author":[{"given":"R","family":"Sergazinov","sequence":"first","affiliation":[{"name":"Department of Statistics, Texas A&M University , 3143 TAMU, 155 Ireland Street , College Station, Texas 77843, \u00a0 mrsergazinov@tamu.edu","place":["U.S.A."]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A","family":"Taeb","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Washington , Box 354322, Seattle, Washington 98195, \u00a0 ataeb@uw.edu","place":["U.S.A."]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4116-0268","authenticated-orcid":false,"given":"I","family":"Gaynanova","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, University of Michigan , 1415 Washington Heights , Ann Arbor, Michigan 48109,","place":["U.S.A"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2025,12,5]]},"reference":[{"key":"2026041720164057300_asaf088-B1","doi-asserted-by":"publisher","author":"Bjoerck","year":"1973","DOI":"10.2307\/2005662"},{"key":"2026041720164057300_asaf088-B2","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1140\/epjb\/e2006-00204-0","article-title":"Large dimension forecasting models and random singular value spectra","volume":"55","author":"Bouchaud","year":"2007","journal-title":"Eur. 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