  In the pooled data problem, the goal is to identify the categories associated
with a large collection of items via a sequence of pooled tests. Each pooled
test reveals the number of items of each category within the pool. We study an
approximate message passing (AMP) algorithm for estimating the categories and
rigorously characterize its performance, in both the noiseless and noisy
settings. For the noiseless setting, we show that the AMP algorithm is
equivalent to one recently proposed by El Alaoui et al. Our results provide a
rigorous version of their performance guarantees, previously obtained via
non-rigorous techniques. For the case of pooled data with two categories, known
as quantitative group testing (QGT), we use the AMP guarantees to compute
precise limiting values of the false positive rate and the false negative rate.
Though the pooled data problem and QGT are both instances of estimation in a
linear model, existing AMP theory cannot be directly applied since the design
matrices are binary valued. The key technical ingredient in our analysis is a
rigorous asymptotic characterization of AMP for generalized linear models
defined via generalized white noise design matrices. This result, established
using a recent universality result of Wang et al., is of independent interest.
Our theoretical results are validated by numerical simulations. For comparison,
we propose estimators based on convex relaxation and iterative thresholding,
without providing theoretical guarantees. The simulations indicate that AMP
outperforms the convex estimator for noiseless pooled data and QGT, but the
convex estimator performs slightly better for noisy pooled data with three
categories when the number of observations is small.
