Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models

Abstract We study the problem of recovering an unknown signal $${\varvec{x}}$$ x given measurements obtained from a generalized linear model with Gaussian sensing matrix. Two popular solutions are based on estimator $$\hat{\varvec{x}}^\mathrm{L}$$ ^ L and spectral $$\hat{\varvec{x}}^\mathrm{s}$$ s . The former is data-dependent combination columns measurement matrix, its analysis quite simple. latter principal eigenvector recent line work has studied performance. In this paper, we show how to optimally combine At heart our exact characterization empirical joint distribution $$({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})$$ ( , ) in high-dimensional limit. This allows us compute Bayes-optimal , limiting When Gaussian, then form $$\theta \hat{\varvec{x}}^\mathrm{L}+\hat{\varvec{x}}^\mathrm{s}$$ θ + derive optimal coefficient. order establish design analyze approximate message passing algorithm whose iterates give approach Numerical simulations demonstrate improvement proposed respect two methods considered separately.