Near optimal efficient decoding from pooled data

The objective of the pooled data problem is to design a measurement matrix $A$ that allows to recover a signal $\SIGMA \in \cbc{0, 1, 2, \ldots, d}^n$ from the observation of the vector $\hat \SIGMA = A \SIGMA$ of joint linear measurements of its components as well as from $A$ itself, using as few measurements as possible. It is both a generalisation of the compelling quantitative group testing problem as well as a special case of the extensively studied compressed sensing problem. If the signal vector is sparse, that is, its number $k$ of non-zero components is much smaller than $n$, it is known that exponential-time constructions to recover $\SIGMA$ from the pair $(A, \hat\SIGMA)$ with no more than $O(k)$ measurements exist. However, so far, all known efficient constructions required at least $\Omega(k\ln n)$ measurements, and it was an open question whether this gap is artificial or of a fundamental nature. In this article we show that indeed, the previous gap between the information-theoretic and computational bounds is not inherent to the problem by providing an efficient recovery algorithm that succeeds with high probability and employs no more than $O(k)$ measurements.