Decoding from Pooled Data: Sharp Information-Theoretic Bounds

TL;DR

This paper establishes sharp information-theoretic bounds on the number of pooled data queries needed to uniquely identify individual types in a population, using probabilistic analysis of a random constraint satisfaction problem.

Contribution

It provides the first rigorous bounds on query complexity for decoding individual data from pooled histograms in a dense, random setting, linking it to planted CSPs and free energy calculations.

Findings

Derived almost matching upper and lower bounds on the number of queries needed.

Connected the problem to the computation of the annealed free energy in the thermodynamic limit.

Discovered a novel identity relating Gaussian integrals over Eulerian flows to spanning tree polynomials.

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.