%0 Unpublished work
%T Decoding from Pooled Data: Phase Transitions of Message Passing
%+ Department of Electrical Engineering and Computer Sciences (Berkeley EECS)
%+ University of California [Berkeley] (UC Berkeley)
%+ Laboratoire de Physique Statistique de l'ENS (LPS)
%+ Institut de Physique ThÃ©orique - UMR CNRS 3681 (IPHT)
%A Alaoui, Ahmed El
%A Ramdas, Aaditya
%A Krzakala, Florent
%A Zdeborova, Lenka
%A Jordan, Michael I.
%Z t17/109
%8 2017-07-03
%D 2017
%Z 1702.02279
%Z Mathematics [math]/Information Theory [math.IT]
%Z Computer Science [cs]/Data Structures and Algorithms [cs.DS]Preprints, Working Papers, ...
%X We consider the problem of decoding a discrete signal of categorical variables from the observation of several histograms of pooled subsets of it. We present an Approximate Message Passing (AMP) algorithm for recovering the signal in the random dense setting where each observed histogram involves a random subset of entries of size proportional to n. We characterize the performance of the algorithm in the asymptotic regime where the number of observations $m$ tends to infinity proportionally to n, by deriving the corresponding State Evolution (SE) equations and studying their dynamics. We initiate the analysis of the multi-dimensional SE dynamics by proving their convergence to a fixed point, along with some further properties of the iterates. The analysis reveals sharp phase transition phenomena where the behavior of AMP changes from exact recovery to weak correlation with the signal as m/n crosses a threshold. We derive formulae for the threshold in some special cases and show that they accurately match experimental behavior.
%G English
%2 https://cea.hal.science/cea-01553606/document
%2 https://cea.hal.science/cea-01553606/file/1702.02279.pdf
%L cea-01553606
%U https://cea.hal.science/cea-01553606
%~ CEA
%~ UNIV-PARIS7
%~ ENS-PARIS
%~ UPMC
%~ LPS
%~ CNRS
%~ INSMI
%~ OPENAIRE
%~ DSM-IPHT
%~ CEA-UPSAY
%~ PSL
%~ USPC
%~ UNIV-PARIS-SACLAY
%~ CEA-UPSAY-SACLAY
%~ UPMC_POLE_2
%~ CEA-DRF
%~ SORBONNE-UNIVERSITE
%~ SU-SCIENCES
%~ UNIV-PARIS
%~ UP-SCIENCES
%~ TEST-HALCNRS
%~ ENS-PSL
%~ SU-TI
%~ GS-MATHEMATIQUES
%~ GS-PHYSIQUE
%~ ALLIANCE-SU