# On the Universality of Noiseless Linear Estimation with Respect to the Measurement Matrix

## TL;DR

This paper demonstrates that the universality of noiseless linear estimation extends beyond random i.i.d. measurement matrices to structured matrices, including the optimal Bayesian reconstruction, using message passing methods.

## Contribution

It shows that universality applies to the Bayes-optimal MMSE error and structured matrices, broadening the understanding of noiseless linear estimation.

## Key findings

- Universality extends to Bayes-optimal MMSE error.
- Universality applies to a range of structured matrices.
- Message passing methods reveal these universal properties.

## Abstract

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.