Novel Impossibility Results for Group-Testing
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In this work we prove non-trivial impossibility results for perhaps the simplest non-linear estimation problem, that of 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 = δ n for any constant δ >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 first that improve over the classical counting lower bound, t/n ≥ H(δ), where H(·) is the binary entropy function. As corollaries of our result, we show that (i) for δ≳ 0.347, individual testing is essentially optimal, i.e., t ≥ n(1-o(1)); and (ii) there is an adaptivity gap, since for δ∈ (0.3471,0.3819) known adaptive GT algorithms require fewer than n tests to reconstruct 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.
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