{"has_accepted_license":"1","date_created":"2021-11-03T10:59:08Z","citation":{"apa":"Mondelli, M., Thrampoulidis, C., &#38; Venkataramanan, R. (2022). 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>","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, 2022. <a href=\"https://doi.org/10.1007/s10208-021-09531-x\">https://doi.org/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>, vol. 22, no. 5, Springer, 2022, pp. 1513–66, doi:<a href=\"https://doi.org/10.1007/s10208-021-09531-x\">10.1007/s10208-021-09531-x</a>.","ama":"Mondelli M, Thrampoulidis C, Venkataramanan R. Optimal combination of linear and spectral estimators for generalized linear models. <i>Foundations of Computational Mathematics</i>. 2022;22(5):1513-1566. 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>, vol. 22, no. 5. Springer, pp. 1513–1566, 2022.","short":"M. Mondelli, C. Thrampoulidis, R. Venkataramanan, Foundations of Computational Mathematics 22 (2022) 1513–1566.","ista":"Mondelli M, Thrampoulidis C, Venkataramanan R. 2022. Optimal combination of linear and spectral estimators for generalized linear models. Foundations of Computational Mathematics. 22(5), 1513–1566."},"type":"journal_article","title":"Optimal combination of linear and spectral estimators for generalized linear models","month":"10","date_updated":"2024-05-22T09:05:47Z","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.","isi":1,"article_processing_charge":"Yes (via OA deal)","keyword":["Applied Mathematics","Computational Theory and Mathematics","Computational Mathematics","Analysis"],"scopus_import":"1","publisher":"Springer","issue":"5","ddc":["510"],"year":"2022","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."}],"publication_identifier":{"eissn":["1615-3383"],"issn":["1615-3375"]},"oa_version":"Published Version","department":[{"_id":"MaMo"}],"status":"public","article_type":"original","date_published":"2022-10-01T00:00:00Z","publication":"Foundations of Computational Mathematics","publication_status":"published","day":"01","doi":"10.1007/s10208-021-09531-x","author":[{"id":"27EB676C-8706-11E9-9510-7717E6697425","orcid":"0000-0002-3242-7020","full_name":"Mondelli, Marco","last_name":"Mondelli","first_name":"Marco"},{"first_name":"Christos","full_name":"Thrampoulidis, Christos","last_name":"Thrampoulidis"},{"first_name":"Ramji","last_name":"Venkataramanan","full_name":"Venkataramanan, Ramji"}],"volume":22,"_id":"10211","external_id":{"arxiv":["2008.03326"],"isi":["000685721000001"]},"project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"intvolume":"        22","quality_controlled":"1","language":[{"iso":"eng"}],"oa":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"page":"1513-1566","file_date_updated":"2021-12-13T15:47:54Z","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","file":[{"file_size":2305731,"date_updated":"2021-12-13T15:47:54Z","content_type":"application/pdf","creator":"alisjak","file_id":"10542","checksum":"9ea12dd8045a0678000a3a59295221cb","success":1,"date_created":"2021-12-13T15:47:54Z","file_name":"2021_Springer_Mondelli.pdf","access_level":"open_access","relation":"main_file"}]}