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We study combinatorial group testing schemes for learning -sparse boolean vectors using highly unreliable disjunctive measurements. We consider an adversarial noise model that only limits the number of false observations, and show that any noise-resilient scheme in this model can only approximately reconstruct the sparse vector. On the positive side, we give a general framework for construction of highly noise-resilient group testing schemes using randomness condensers. Simple randomized instantiations of this construction give non-adaptive measurement schemes, with measurements, that allow efficient reconstruction of -sparse vectors up to false positives even in the presence of false positives and false negatives within the measurement outcomes, for any constant . None of these parameters can be substantially improved without dramatically affecting the others. Furthermore, we obtain several explicit (and incomparable) constructions, in particular one matching the randomized trade-off but using measurements. We also obtain explicit constructions that allow fast reconstruction in time , which would be sublinear in for sufficiently sparse vectors.
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Luca Baldassarre, Nirav Bhan, Volkan Cevher