Abstract
Consider 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 tends to infinity. 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.
Highlights
The theory of compressed sensing, where one is interested in recovering a high-dimensional signal from a small number of measurements, has grown into a rich field of investigation and found many applications [24]
That is, letting \scrZ be the number of solutions of the constraint satisfaction problem (CSP), we show that the limit
It may seem plausible that the upper bound is loose due to a possible lack of concentration of the random variable \scrZ about its mean, and this translates to the possibility of existence of a nontrivial interval inside [\gama \mathrm{\mathrm{\mathrm{, \gama \mathrm{\mathrm{] where \scrZ is typically close to 1 while its expectation is exponentially large. This is a standard issue in the use of the first moment method encountered in many random CSPs
Summary
The theory of compressed sensing, where one is interested in recovering a high-dimensional signal from a small number of measurements, has grown into a rich field of investigation and found many applications [24]. We attempt to fill the information-theoretic gap in the random setting by providing tighter upper and lower bounds on the number of queries m necessary and sufficient to uniquely determine the planted assignment \tau \ast with high probability, which depend on the dimension d and \bfitpi along with explicit constants.
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