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The pooled data problem asks to identify the unknown labels of a set of itemsfrom condensed measurements. More precisely, given $n$ items, assume that eachitem has a label in $\cbc{0,1,\ldots, d}$, encoded via the ground-truth$\SIGMA$. We call the pooled data problem sparse if the number of non-zeroentries of $\SIGMA$ scales as $k \sim n^{\theta}$ for $\theta \in (0,1)$. Theinformation that is revealed about $\SIGMA$ comes from pooled measurements,each indicating how many items of each label are contained in the pool. Themost basic question is to design a pooling scheme that uses as few pools aspossible, while reconstructing $\SIGMA$ with high probability. Variants of theproblem and its combinatorial ramifications have been studied for at least 35years. However, the study of the modern question of \emph{efficient} inferenceof the labels has suggested a statistical-to-computational gap of order $\logn$ in the minimum number of pools needed for theoretically possible versusefficient inference. In this article, we resolve the question whether this$\log n$-gap is artificial or of a fundamental nature by the design of anefficient algorithm, called \algoname, based upon a novel pooling scheme on anumber of pools very close to the information-theoretic threshold.

On a Near-Optimal \& Efficient Algorithm for the Sparse Pooled Data Problem