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

Abstract : In a noiseless linear estimation problem, one aims to reconstruct a vector x * from the knowledge of its linear projections y = Φx *. There have been many theoretical works concentrating on the case where the matrix Φ 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 1 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.
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Submitted on : Wednesday, April 1, 2020 - 4:56:05 PM
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Alia Abbara, Antoine Baker, Florent Krzakala, Lenka Zdeborová. On the universality of noiseless linear estimation with respect to the measurement matrix. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2020, 53 (16), pp.164001. ⟨10.1088/1751-8121/ab59ef⟩. ⟨cea-02528193⟩

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