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Two popular solutions are based on a linear estimator $$\\hat{\\varvec{x}}^\\mathrm{L}$$<\/jats:tex-math>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n L<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and a spectral estimator $$\\hat{\\varvec{x}}^\\mathrm{s}$$<\/jats:tex-math>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. 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 $$\\hat{\\varvec{x}}^\\mathrm{L}$$<\/jats:tex-math>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n L<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\hat{\\varvec{x}}^\\mathrm{s}$$<\/jats:tex-math>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. At the heart of our analysis is the exact characterization of the empirical joint distribution of $$({\\varvec{x}}, \\hat{\\varvec{x}}^\\mathrm{L}, \\hat{\\varvec{x}}^\\mathrm{s})$$<\/jats:tex-math>\n \n (<\/mml:mo>\n \n x<\/mml:mi>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n L<\/mml:mi>\n <\/mml:msup>\n ,<\/mml:mo>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n s<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of $$\\hat{\\varvec{x}}^\\mathrm{L}$$<\/jats:tex-math>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n L<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\hat{\\varvec{x}}^\\mathrm{s}$$<\/jats:tex-math>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, given the limiting distribution of the signal $${\\varvec{x}}$$<\/jats:tex-math>\n \n x<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form $$\\theta \\hat{\\varvec{x}}^\\mathrm{L}+\\hat{\\varvec{x}}^\\mathrm{s}$$<\/jats:tex-math>\n \n \u03b8<\/mml:mi>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n L<\/mml:mi>\n <\/mml:msup>\n +<\/mml:mo>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and we derive the optimal combination coefficient. In order to establish the limiting distribution of $$({\\varvec{x}}, \\hat{\\varvec{x}}^\\mathrm{L}, \\hat{\\varvec{x}}^\\mathrm{s})$$<\/jats:tex-math>\n \n (<\/mml:mo>\n \n x<\/mml:mi>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n L<\/mml:mi>\n <\/mml:msup>\n ,<\/mml:mo>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n s<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we design and analyze an approximate message passing algorithm whose iterates give $$\\hat{\\varvec{x}}^\\mathrm{L}$$<\/jats:tex-math>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n L<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and approach $$\\hat{\\varvec{x}}^\\mathrm{s}$$<\/jats:tex-math>\n \n \n \n x<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.<\/jats:p>","DOI":"10.1007\/s10208-021-09531-x","type":"journal-article","created":{"date-parts":[[2021,8,17]],"date-time":"2021-08-17T18:02:54Z","timestamp":1629223374000},"page":"1513-1566","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models"],"prefix":"10.1007","volume":"22","author":[{"given":"Marco","family":"Mondelli","sequence":"first","affiliation":[]},{"given":"Christos","family":"Thrampoulidis","sequence":"additional","affiliation":[]},{"given":"Ramji","family":"Venkataramanan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,8,17]]},"reference":[{"key":"9531_CR1","doi-asserted-by":"crossref","unstructured":"Bahmani, S., Romberg, J.: Phase retrieval meets statistical learning theory: A flexible convex relaxation. 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