https://hal-cea.archives-ouvertes.fr/cea-01448031Alaoui, Ahmed ElAhmed ElAlaouiLBNL - Lawrence Berkeley National Laboratory [Berkeley]Ramdas, AadityaAadityaRamdasLBNL - Lawrence Berkeley National Laboratory [Berkeley]Krzakala, FlorentFlorentKrzakalaLPS - Laboratoire de Physique Statistique de l'ENS - FRDPENS - Fédération de recherche du Département de physique de l'Ecole Normale Supérieure - ENS Paris - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris sciences et lettres - CNRS - Centre National de la Recherche Scientifique - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche ScientifiqueZdeborova, LenkaLenkaZdeborovaIPHT - Institut de Physique Théorique - UMR CNRS 3681 - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - Université Paris-Saclay - CNRS - Centre National de la Recherche ScientifiqueJordan, Michael I.Michael I.JordanLBNL - Lawrence Berkeley National Laboratory [Berkeley]Decoding from Pooled Data: Sharp Information-Theoretic BoundsHAL CCSD2017[MATH.MATH-PR] Mathematics [math]/Probability [math.PR][MATH.MATH-IT] Mathematics [math]/Information Theory [math.IT]De Laborderie, EmmanuelleStatistical Physics Approach to Reconstruction in Compressed Sensing - SPARCS - - EC:FP7:ERC2012-10-01 - 2017-09-30 - 307087 - VALID - 2017-01-27 14:42:382023-03-22 12:24:432017-01-27 16:17:53enPreprints, Working Papers, ...https://hal-cea.archives-ouvertes.fr/cea-01448031/documenttext/html; charset=utf-81Consider a population consisting of n individuals, each of whom has one of d types (e.g. their blood type, in which case d = 4). We are allowed to query this database by specifying a subset of the population, and in response we observe a noiseless histogram (a d-dimensional vector of counts) of types of the pooled individuals. This measurement model arises in practical situations such as pooling of genetic data and may also be motivated by privacy considerations. We are interested in the number of queries one needs to unambiguously determine the type of each individual. In this paper, we study this information-theoretic question under the random, dense setting where in each query, a random subset of individuals of size proportional to n is chosen. This makes the problem a particular example of a random constraint satisfaction problem (CSP) with a " planted " solution. We establish almost matching upper and lower bounds on the minimum number of queries m such that there is no solution other than the planted one with probability tending to 1 as n → ∞. Our proof relies on the computation of the exact " annealed free energy " of this model in the thermodynamic limit, which corresponds to the exponential rate of decay of the expected number of solution to this planted CSP. As a by-product of the analysis, we show an identity of independent interest relating the Gaussian integral over the space of Eulerian flows of a graph to its spanning tree polynomial.