article published yes Marco Mondelli author 27EB676C-8706-11E9-9510-7717E66974250000-0002-3242-7020 Christos Thrampoulidis author Ramji Venkataramanan author MaMo department IST Austria Open Access Fund project 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. https://research-explorer.ista.ac.at/download/10211/10542/2021_Springer_Mondelli.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Springer2022 eng Applied MathematicsComputational Theory and MathematicsComputational MathematicsAnalysis 1615-3375 1615-3383 2008.03326 00068572100000110.1007/s10208-021-09531-x 2251513-1566 M. Mondelli, C. Thrampoulidis, R. Venkataramanan, Foundations of Computational Mathematics 22 (2022) 1513–1566. 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> 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>. 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> 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>. 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. 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. 102112021-11-03T10:59:08Z2025-04-15T06:53:08Z