{"author":[{"last_name":"Mondelli","full_name":"Mondelli, Marco","first_name":"Marco","id":"27EB676C-8706-11E9-9510-7717E6697425","orcid":"0000-0002-3242-7020"},{"full_name":"Thrampoulidis, Christos","first_name":"Christos","last_name":"Thrampoulidis"},{"last_name":"Venkataramanan","first_name":"Ramji","full_name":"Venkataramanan, Ramji"}],"article_processing_charge":"Yes (via OA deal)","oa":1,"oa_version":"Published Version","doi":"10.1007/s10208-021-09531-x","department":[{"_id":"MaMo"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","publication_identifier":{"issn":["1615-3375"],"eissn":["1615-3383"]},"abstract":[{"lang":"eng","text":"We study the problem of recovering an unknown signal 𝑥𝑥 given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator 𝑥𝑥^L and a spectral estimator 𝑥𝑥^s. The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine 𝑥𝑥^L and 𝑥𝑥^s. At the heart of our analysis is the exact characterization of the empirical joint distribution of (𝑥𝑥,𝑥𝑥^L,𝑥𝑥^s) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of 𝑥𝑥^L and 𝑥𝑥^s, given the limiting distribution of the signal 𝑥𝑥. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form 𝜃𝑥𝑥^L+𝑥𝑥^s and we derive the optimal combination coefficient. In order to establish the limiting distribution of (𝑥𝑥,𝑥𝑥^L,𝑥𝑥^s), we design and analyze an approximate message passing algorithm whose iterates give 𝑥𝑥^L and approach 𝑥𝑥^s. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately."}],"date_published":"2021-08-17T00:00:00Z","date_updated":"2023-09-05T14:13:57Z","_id":"10211","external_id":{"arxiv":["2008.03326"],"isi":["000685721000001"]},"publication":"Foundations of Computational Mathematics","status":"public","article_type":"original","scopus_import":"1","day":"17","isi":1,"type":"journal_article","publisher":"Springer","month":"08","publication_status":"published","language":[{"iso":"eng"}],"acknowledgement":"M. Mondelli would like to thank Andrea Montanari for helpful discussions. All the authors would like to thank the anonymous reviewers for their helpful comments.","quality_controlled":"1","title":"Optimal combination of linear and spectral estimators for generalized linear models","file":[{"content_type":"application/pdf","file_id":"10542","file_size":2305731,"date_updated":"2021-12-13T15:47:54Z","access_level":"open_access","date_created":"2021-12-13T15:47:54Z","checksum":"9ea12dd8045a0678000a3a59295221cb","relation":"main_file","success":1,"file_name":"2021_Springer_Mondelli.pdf","creator":"alisjak"}],"ddc":["510"],"date_created":"2021-11-03T10:59:08Z","has_accepted_license":"1","year":"2021","citation":{"short":"M. Mondelli, C. Thrampoulidis, R. Venkataramanan, Foundations of Computational Mathematics (2021).","ama":"Mondelli M, Thrampoulidis C, Venkataramanan R. Optimal combination of linear and spectral estimators for generalized linear models. <i>Foundations of Computational Mathematics</i>. 2021. doi:<a href=\"https://doi.org/10.1007/s10208-021-09531-x\">10.1007/s10208-021-09531-x</a>","mla":"Mondelli, Marco, et al. “Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models.” <i>Foundations of Computational Mathematics</i>, Springer, 2021, doi:<a href=\"https://doi.org/10.1007/s10208-021-09531-x\">10.1007/s10208-021-09531-x</a>.","ieee":"M. Mondelli, C. Thrampoulidis, and R. Venkataramanan, “Optimal combination of linear and spectral estimators for generalized linear models,” <i>Foundations of Computational Mathematics</i>. Springer, 2021.","apa":"Mondelli, M., Thrampoulidis, C., &#38; Venkataramanan, R. (2021). Optimal combination of linear and spectral estimators for generalized linear models. <i>Foundations of Computational Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s10208-021-09531-x\">https://doi.org/10.1007/s10208-021-09531-x</a>","ista":"Mondelli M, Thrampoulidis C, Venkataramanan R. 2021. Optimal combination of linear and spectral estimators for generalized linear models. Foundations of Computational Mathematics.","chicago":"Mondelli, Marco, Christos Thrampoulidis, and Ramji Venkataramanan. “Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models.” <i>Foundations of Computational Mathematics</i>. Springer, 2021. <a href=\"https://doi.org/10.1007/s10208-021-09531-x\">https://doi.org/10.1007/s10208-021-09531-x</a>."},"file_date_updated":"2021-12-13T15:47:54Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"keyword":["Applied Mathematics","Computational Theory and Mathematics","Computational Mathematics","Analysis"]}