Quantitative Group Testing and Pooled Data in the Linear Regime With Sublinear Tests

In the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pooled data</i> 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 in the pool belonging to each category. A prominent special case is quantitative group testing (QGT), which is the case of pooled data with two categories. We consider these problems in the non-adaptive and linear regime, where the fraction of items in each category is of constant order. We propose a scheme with a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spatially coupled</i> Bernoulli test matrix and an efficient approximate message passing (AMP) algorithm for recovery. We rigorously characterize its asymptotic performance in both the noiseless and noisy settings, and prove that in the noiseless case, the AMP algorithm achieves <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">almost-exact</i> recovery with a number of tests sublinear in the total number of items <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>. Although there exist other efficient schemes for noiseless QGT and pooled data that achieve recovery with order-optimal sample complexity ((Θ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>/log <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> ) tests, there are no guarantees on their performance in the presence of noise, even at low noise-levels. In comparison, our scheme achieves recovery in the noiseless case with a number of tests sublinear in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>, and its performance degrades gracefully in the presence of noise. Numerical simulations illustrate the benefits of the spatially coupled scheme at finite dimensions, showing that it outperforms i.i.d. test designs as well as other recovery algorithms based on convex programming.