On the universality of noiseless linear estimation with respect to the measurement matrix

Abstract

In a noiseless linear estimation problem, the goal is to reconstruct a vector from the knowledge of its linear projections. There have been many theoretical works concentrating on the case where the matrix is a random i.i.d. one, but a number of heuristic evidence suggests that many of these results are universal and extend well beyond this restricted case. Here we revisit this problem through the prism of development of message passing methods, and consider not only the universality of the transition, as previously addressed, but also the one of the optimal Bayesian reconstruction. We observed that the universality extends to the Bayes-optimal minimum mean-squared (MMSE) error, and to a range of structured matrices.