Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models

We study the problem of recovering an unknown signal boldsymbol x given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator hatboldsymbol xrm L and a spectral estimator hatboldsymbol xrm 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 hatboldsymbol xrm L and hatboldsymbol xrm s. At the heart of our analysis is the exact characterization of the joint empirical distribution of (boldsymbol x, hatboldsymbol xrm L, hatboldsymbol xrm s) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of hatboldsymbol xrm L and hatboldsymbol xrm s, given the limiting distribution of the signal boldsymbol x. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form θhatboldsymbol xrm L+hatboldsymbol xrm s and we derive the optimal combination coefficient. In order to establish the limiting distribution of (boldsymbol x, hatboldsymbol xrm L, hatboldsymbol xrm s), we design and analyze an Approximate Message Passing (AMP) algorithm whose iterates give hatboldsymbol xrm L and approach hatboldsymbol xrm s. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.