@article{10211,
  abstract     = {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.},
  author       = {Mondelli, Marco and Thrampoulidis, Christos and Venkataramanan, Ramji},
  issn         = {1615-3383},
  journal      = {Foundations of Computational Mathematics},
  keywords     = {Applied Mathematics, Computational Theory and Mathematics, Computational Mathematics, Analysis},
  number       = {5},
  pages        = {1513--1566},
  publisher    = {Springer},
  title        = {{Optimal combination of linear and spectral estimators for generalized linear models}},
  doi          = {10.1007/s10208-021-09531-x},
  volume       = {22},
  year         = {2022},
}