{"project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"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.","department":[{"_id":"MaMo"}],"doi":"10.1007/s10208-021-09531-x","abstract":[{"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.","lang":"eng"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"fulldoi":"https://doi.org/10.1007/s10208-021-09531-x","scopus_import":"1","publication_status":"published","external_id":{"isi":["000685721000001"],"arxiv":["2008.03326"]},"intvolume":"        22","citation":{"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>.","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.","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>","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>","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.","short":"M. Mondelli, C. Thrampoulidis, R. Venkataramanan, Foundations of Computational Mathematics 22 (2022) 1513–1566.","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>."},"date_published":"2022-10-01T00:00:00Z","title":"Optimal combination of linear and spectral estimators for generalized linear models","arxiv":1,"type":"journal_article","month":"10","status":"public","license":"https://creativecommons.org/licenses/by/4.0/","page":"1513-1566","year":"2022","keyword":["Applied Mathematics","Computational Theory and Mathematics","Computational Mathematics","Analysis"],"issue":"5","publication_identifier":{"issn":["1615-3375"],"eissn":["1615-3383"]},"volume":22,"date_updated":"2025-04-15T06:53:08Z","file_date_updated":"2021-12-13T15:47:54Z","author":[{"last_name":"Mondelli","id":"27EB676C-8706-11E9-9510-7717E6697425","full_name":"Mondelli, Marco","orcid":"0000-0002-3242-7020","first_name":"Marco"},{"last_name":"Thrampoulidis","full_name":"Thrampoulidis, Christos","first_name":"Christos"},{"full_name":"Venkataramanan, Ramji","first_name":"Ramji","last_name":"Venkataramanan"}],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"article_type":"original","isi":1,"file":[{"file_name":"2021_Springer_Mondelli.pdf","date_created":"2021-12-13T15:47:54Z","relation":"main_file","content_type":"application/pdf","date_updated":"2021-12-13T15:47:54Z","success":1,"file_id":"10542","access_level":"open_access","file_size":2305731,"creator":"alisjak","checksum":"9ea12dd8045a0678000a3a59295221cb"}],"date_created":"2021-11-03T10:59:08Z","quality_controlled":"1","oa":1,"has_accepted_license":"1","_id":"10211","oa_version":"Published Version","publisher":"Springer","article_processing_charge":"Yes (via OA deal)","publication":"Foundations of Computational Mathematics","day":"01","language":[{"iso":"eng"}]}