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.
翻译:集合数据问题的目标是设计一个测量矩阵,用尽可能少的测量方法,用尽可能少的测量方法,设计一个测量矩阵,以便从对矢量的观测中恢复一个信号$SIGMA=ccc{0,1,2,eldots,d ⁇ n$,从对矢量的观测中恢复一个信号$SIGMA=SIGMA=A\SIGMA美元,以及从对其组件进行联合线性测量的美元中恢复一个信号$SIGMA美元,同时使用尽可能少的测量方法。它既是对令人信服的定量组量测试问题的概括,也是广泛研究压缩感测问题的特例。如果信号矢量是稀少的,即其非零度部件的美元数比美元少得多,那么,如果其非零度的测量方法的数值比美元要小得多,那么已知要从对一对一对美元(A,hat\s\SIGMA)的指数性测量方法来收回美元,那么,我们确实要用一个不难的问题是,因为先前的回收概率和高的计算方法,我们并没有显示以前的一个差距。