[{"year":"2026","ddc":["000"],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"has_accepted_license":"1","month":"06","mathsc":["62E20","62J05","62J12"],"date_published":"2026-06-01T00:00:00Z","_id":"22228","quality_controlled":"1","date_updated":"2026-07-01T06:29:52Z","language":[{"iso":"eng"}],"volume":8,"title":"Precise asymptotics for spectral methods in mixed generalized linear models","status":"public","page":"411-439","PlanS_conform":"1","corr_author":"1","arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2026-06-30T13:03:41Z","article_type":"original","publisher":"Society for Industrial & Applied Mathematics","publication_status":"published","publication":"SIAM Journal on Mathematics of Data Science","file":[{"access_level":"open_access","relation":"main_file","content_type":"application/pdf","date_created":"2026-07-01T06:22:15Z","file_size":1210346,"date_updated":"2026-07-01T06:22:15Z","file_id":"22230","creator":"dernst","file_name":"2026_SIAMJourmathDataScience_Zhang.pdf","success":1,"checksum":"5cfd350dc64d1476063e959316dbff65"}],"abstract":[{"lang":"eng","text":"In a mixed generalized linear model, the goal is to learn multiple signals from unlabeled observations: each sample comes from exactly one signal, but it is not known which one. We consider the prototypical problem of estimating two statistically independent signals in a mixed generalized linear model with Gaussian covariates. Spectral methods are a popular class of estimators which output the top two eigenvectors of a suitable data-dependent matrix. However, despite the wide applicability, their design is still obtained via heuristic considerations, and the number of samples ๐ needed to guarantee recovery is superlinear in the signal dimension ๐. In this paper, we develop exact asymptotics on spectral methods in the challenging proportional regime in which ๐,๐ grow large and their ratio converges to a finite constant. This allows us optimize the design of the spectral method, and combine it with a simple linear estimator, to minimize the estimation error. Our characterization exploits a mix of tools from random matrices, free probability, and the theory of approximate message passing algorithms. Numerical simulations for mixed linear regression and phase retrieval demonstrate the advantage enabled by our analysis over existing designs of spectral methods."}],"doi":"10.1137/24m1702854","oa_version":"Published Version","publication_identifier":{"eissn":["2577-0187"]},"researchdata_availability":"no","acknowledgement":"The first and second authors were partially supported by the 2019 Lopez-Loreta prize.","department":[{"_id":"MaMo"}],"issue":"2","project":[{"_id":"059876FA-7A3F-11EA-A408-12923DDC885E","name":"Prix Lopez-Loretta 2019 - Marco Mondelli"}],"author":[{"last_name":"Zhang","full_name":"Zhang, Yihan","first_name":"Yihan"},{"id":"27EB676C-8706-11E9-9510-7717E6697425","first_name":"Marco","orcid":"0000-0002-3242-7020","full_name":"Mondelli, Marco","last_name":"Mondelli"},{"first_name":"Ramji","last_name":"Venkataramanan","full_name":"Venkataramanan, Ramji"}],"OA_type":"hybrid","external_id":{"arxiv":["2211.11368"]},"supplementarymaterial":"no","scopus_import":"1","intvolume":" 8","das_tickbox":"0","file_date_updated":"2026-07-01T06:22:15Z","OA_place":"publisher","keyword":["spectral estimator","generalized linear models","mixed regression","high-dimensional asymptotics","random matrix theory","approximate message passing (AMP)"],"day":"01","citation":{"mla":"Zhang, Yihan, et al. โPrecise Asymptotics for Spectral Methods in Mixed Generalized Linear Models.โ SIAM Journal on Mathematics of Data Science, vol. 8, no. 2, Society for Industrial & Applied Mathematics, 2026, pp. 411โ39, doi:10.1137/24m1702854.","ista":"Zhang Y, Mondelli M, Venkataramanan R. 2026. Precise asymptotics for spectral methods in mixed generalized linear models. SIAM Journal on Mathematics of Data Science. 8(2), 411โ439.","apa":"Zhang, Y., Mondelli, M., & Venkataramanan, R. (2026). Precise asymptotics for spectral methods in mixed generalized linear models. SIAM Journal on Mathematics of Data Science. Society for Industrial & Applied Mathematics. https://doi.org/10.1137/24m1702854","ieee":"Y. Zhang, M. Mondelli, and R. Venkataramanan, โPrecise asymptotics for spectral methods in mixed generalized linear models,โ SIAM Journal on Mathematics of Data Science, vol. 8, no. 2. Society for Industrial & Applied Mathematics, pp. 411โ439, 2026.","short":"Y. Zhang, M. Mondelli, R. Venkataramanan, SIAM Journal on Mathematics of Data Science 8 (2026) 411โ439.","ama":"Zhang Y, Mondelli M, Venkataramanan R. Precise asymptotics for spectral methods in mixed generalized linear models. SIAM Journal on Mathematics of Data Science. 2026;8(2):411-439. doi:10.1137/24m1702854","chicago":"Zhang, Yihan, Marco Mondelli, and Ramji Venkataramanan. โPrecise Asymptotics for Spectral Methods in Mixed Generalized Linear Models.โ SIAM Journal on Mathematics of Data Science. Society for Industrial & Applied Mathematics, 2026. https://doi.org/10.1137/24m1702854."},"oa":1,"article_processing_charge":"Yes (in subscription journal)","type":"journal_article"}]