--- res: bibo_abstract: - "We study the problem of recovering an unknown signal \U0001D465\U0001D465 given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator \U0001D465\U0001D465^L and a spectral estimator \U0001D465\U0001D465^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 \U0001D465\U0001D465^L and \U0001D465\U0001D465^s. At the heart of our analysis is the exact characterization of the empirical joint distribution of (\U0001D465\U0001D465,\U0001D465\U0001D465^L,\U0001D465\U0001D465^s) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of \U0001D465\U0001D465^L and \U0001D465\U0001D465^s, given the limiting distribution of the signal \U0001D465\U0001D465. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form \U0001D703\U0001D465\U0001D465^L+\U0001D465\U0001D465^s and we derive the optimal combination coefficient. In order to establish the limiting distribution of (\U0001D465\U0001D465,\U0001D465\U0001D465^L,\U0001D465\U0001D465^s), we design and analyze an approximate message passing algorithm whose iterates give \U0001D465\U0001D465^L and approach \U0001D465\U0001D465^s. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.@eng" bibo_authorlist: - foaf_Person: foaf_givenName: Marco foaf_name: Mondelli, Marco foaf_surname: Mondelli foaf_workInfoHomepage: http://www.librecat.org/personId=27EB676C-8706-11E9-9510-7717E6697425 orcid: 0000-0002-3242-7020 - foaf_Person: foaf_givenName: Christos foaf_name: Thrampoulidis, Christos foaf_surname: Thrampoulidis - foaf_Person: foaf_givenName: Ramji foaf_name: Venkataramanan, Ramji foaf_surname: Venkataramanan bibo_doi: 10.1007/s10208-021-09531-x dct_date: 2021^xs_gYear dct_identifier: - UT:000685721000001 dct_isPartOf: - http://id.crossref.org/issn/1615-3375 - http://id.crossref.org/issn/1615-3383 dct_language: eng dct_publisher: Springer@ dct_subject: - Applied Mathematics - Computational Theory and Mathematics - Computational Mathematics - Analysis dct_title: Optimal combination of linear and spectral estimators for generalized linear models@ ...