We study the problem of recovering an unknown signal x\documentclass[12pt]{minimal}
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\begin{document}$${\varvec{x}}$$\end{document} given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator x^L\documentclass[12pt]{minimal}
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\begin{document}$$\hat{\varvec{x}}^\mathrm{L}$$\end{document} and a spectral estimator x^s\documentclass[12pt]{minimal}
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\begin{document}$$\hat{\varvec{x}}^\mathrm{s}$$\end{document}. 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 x^L\documentclass[12pt]{minimal}
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\begin{document}$$\hat{\varvec{x}}^\mathrm{L}$$\end{document} and x^s\documentclass[12pt]{minimal}
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\begin{document}$$\hat{\varvec{x}}^\mathrm{s}$$\end{document}. At the heart of our analysis is the exact characterization of the empirical joint distribution of (x,x^L,x^s)\documentclass[12pt]{minimal}
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\begin{document}$$({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})$$\end{document} in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of x^L\documentclass[12pt]{minimal}
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\begin{document}$$\hat{\varvec{x}}^\mathrm{L}$$\end{document} and x^s\documentclass[12pt]{minimal}
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\begin{document}$$\hat{\varvec{x}}^\mathrm{s}$$\end{document}, given the limiting distribution of the signal x\documentclass[12pt]{minimal}
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\begin{document}$${\varvec{x}}$$\end{document}. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form θx^L+x^s\documentclass[12pt]{minimal}
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\begin{document}$$\theta \hat{\varvec{x}}^\mathrm{L}+\hat{\varvec{x}}^\mathrm{s}$$\end{document} and we derive the optimal combination coefficient. In order to establish the limiting distribution of (x,x^L,x^s)\documentclass[12pt]{minimal}
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\begin{document}$$({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})$$\end{document}, we design and analyze an approximate message passing algorithm whose iterates give x^L\documentclass[12pt]{minimal}
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\begin{document}$$\hat{\varvec{x}}^\mathrm{L}$$\end{document} and approach x^s\documentclass[12pt]{minimal}
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\begin{document}$$\hat{\varvec{x}}^\mathrm{s}$$\end{document}. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.
