In this paper, we derive mutual information based upper and lower bounds on
the number of nonadaptive group tests required to identify a given number of
"non defective" items from a large population containing a small number of
"defective" items. We show that a reduction in the number of tests is
achievable compared to the approach of first identifying all the defective
items and then picking the required number of non-defective items from the
complement set. In the asymptotic regime with the population size Nββ, to identify L non-defective items out of a population
containing K defective items, when the tests are reliable, our results show
that 1βo(1)CsβKβ(Ξ¦(Ξ±0β,Ξ²0β)+o(1)) measurements are
sufficient, where Csβ is a constant independent of N,K and L, and
Ξ¦(Ξ±0β,Ξ²0β) is a bounded function of Ξ±0ββlimNβββNβKLβ and Ξ²0ββlimNβββNβKKβ. Further, in the nonadaptive group
testing setup, we obtain rigorous upper and lower bounds on the number of tests
under both dilution and additive noise models. Our results are derived using a
general sparse signal model, by virtue of which, they are also applicable to
other important sparse signal based applications such as compressive sensing.Comment: 32 pages, 2 figures, 3 tables, revised version of a paper submitted
to IEEE Trans. Inf. Theor