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Generalized Group Testing
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:10777-10835, 2022.
Abstract
In the problem of classical group testing one aims to identify a small subset (of size d) diseased individuals/defective items in a large population (of size n) via a minimal number of suitably-designed group tests on subsets of items, where the test outcome is positive iff the given test contains at least one defective item. Motivated by physical considerations, we consider a generalized setting that includes as special cases multiple other group-testing-like models in the literature. In our setting, which subsumes as special cases a variety of noiseless and noisy group-testing models in the literature, the test outcome is positive with probability f(x), where x is the number of defectives tested in a pool, and f(⋅) is an arbitrary {\it monotonically increasing} (stochastic) test function. Our main contributions are as follows. 1. We present a non-adaptive scheme that with probability 1−ε identifies all defective items. Our scheme requires at most O(H(f)dlog(n/ε)) tests, where H(f) is a suitably defined “sensitivity parameter" of f(⋅), and is never larger than O(d1+o(1)), but may be substantially smaller for many f(⋅). 2. We argue that any non-adaptive group testing scheme needs at least Ω(h(f)dlog(n/d)) tests to ensure high reliability recovery. Here h(f) is a suitably defined “concentration parameter" of f(⋅), and h(f)∈Ω(1). 3. We prove that our sample-complexity bounds for generalized group testing are information-theoretically near-optimal for a variety of sparse-recovery group-testing models in the literature. That is, for {\it any} “noisy" test function f(⋅) (i.e. 0<f(0)<f(d)<1), and for a variety of “(one-sided) noiseless" test functions f(⋅) (i.e., either f(0)=0, or f(d)=1, or both) studied in the literature we show that H(f)/h(f)∈Θ(1). As a by-product we tightly characterize the heretofore open information-theoretic sample-complexity for the well-studied model of threshold group-testing. For general (near)-noiseless test functions f(⋅) we show that H(f)/h(f)∈O(d1+o(1)). We also demonstrate a “natural" test-function f(⋅) whose sample complexity scales “extremally" as Θ(d2log(n)), rather than Θ(dlog(n)) as in the case of classical group-testing. Some of our techniques may be of independent interest – in particular our achievability requires a delicate saddle-point approximation, and our impossibility proof relies on a novel bound relating the mutual information of pair of random variables with the mean and variance of a specific function, and we derive novel structural results about monotone functions.