On the Universality of Noiseless Linear Estimation with Respect to the Measurement Matrix
This addresses a theoretical gap in signal processing and machine learning by showing that key performance metrics are robust across different matrix designs, which is incremental but important for practical applications.
The paper investigates whether theoretical results for noiseless linear estimation, typically derived for random i.i.d. measurement matrices, extend universally to other matrix types, finding that universality holds for the Bayes-optimal minimum mean-squared error and a range of structured matrices.
In a noiseless linear estimation problem, one aims to reconstruct a vector x* from the knowledge of its linear projections y=Phi x*. There have been many theoretical works concentrating on the case where the matrix Phi 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 problematic through the prism of development of message passing methods, and consider not only the universality of the l1 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.