# Decoding from Pooled Data: Phase Transitions of Message Passing

## TL;DR

This paper introduces an AMP algorithm for decoding categorical signals from pooled histogram data, analyzing its phase transition behavior and establishing thresholds for exact recovery in high-dimensional settings.

## Contribution

It develops a novel AMP-based decoding method for pooled histogram data and characterizes its phase transition thresholds through rigorous state evolution analysis.

## Key findings

- Identifies sharp phase transition thresholds for exact recovery.
- Derives formulae for thresholds that match experimental results.
- Proves convergence properties of the multi-dimensional state evolution.

## Abstract

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.