The fundamental task of group testing is to recover a small distinguished subset of items from a large group while efficiently reducing the total number of tests (measurements). The key contribution of this paper is in adopting a new information-theoretic perspective on group testing problems. Establishing its connection to Shannon-coding theory, we formulate the group testing problem as a channel coding/decoding problem and derive a unifying result that reduces many of the interesting questions to computation of a mutual information expression. First, we derive an achievable bound on the total number of tests using random constructions and typical set decoding. This result is fairly general; it allows us to verify existing bounds for some of the known scenarios and extend the analysis to many new interesting setups including noisy versions of group testing. We obtain tradeoffs between the number of tests, sparsity and the group size for a wide range of models. Among the models we consider are the deterministic noise-free case, approximate reconstruction with bounded distortion, additive measurement noise, and dilution models. Then, we show that the achievable bound provides fairly tight asymptotic scaling by deriving an information theoretic lower-bound using Fano's inequality.
Recently Atia and Saligrama REF considered the problem of noisy group testing from an information theoretic perspective (see reference therein for related work on group testing).
Boolean Compressed Sensing and Noisy Group Testing
IEEE Trans.Inf.Theory 58 (2012) 1880-1901
Computer Science Mathematics