Novel Impossibility Results for Group-Testing

In this work we prove non-trivial impossibility results for perhaps the simplest non-linear estimation problem, that of {\it Group Testing} (GT), via the recently developed Madiman-Tetali inequalities. Group Testing concerns itself with identifying a hidden set of $d$ defective items from a set of $n$ items via $t$ {disjunctive/pooled} measurements ("group tests"). We consider the linear sparsity regime, i.e. $d = \delta n$ for any constant $\delta >0$, a hitherto little-explored (though natural) regime. In a standard information-theoretic setting, where the tests are required to be non-adaptive and a small probability of reconstruction error is allowed, our lower bounds on $t$ are the {\it first} that improve over the classical counting lower bound, $t/n \geq H(\delta)$, where $H(\cdot)$ is the binary entropy function. As corollaries of our result, we show that (i) for $\delta \gtrsim 0.347$, individual testing is essentially optimal, i.e., $t \geq n(1-o(1))$; and (ii) there is an {adaptivity gap}, since for $\delta \in (0.3471,0.3819)$ known {adaptive} GT algorithms require fewer than $n$ tests to reconstruct ${\cal D}$, whereas our bounds imply that the best nonadaptive algorithm must essentially be individual testing of each element. Perhaps most importantly, our work provides a framework for combining combinatorial and information-theoretic methods for deriving non-trivial lower bounds for a variety of non-linear estimation problems.