Generalized Group Testing

In the problem of classical group testing one aims to identify a small subset (of size $d$ ) diseased individuals/defective items in large population notation="LaTeX">$n$ ). This process is based on minimal number suitably-designed tests subsets items, where test outcome positive iff given contains at least defective item. Motivated by physical considerations, such as scenarios with imperfect apparatus, we consider generalized setting that includes special cases multiple other group-testing-like models literature. our governed an arbitrary monotonically increasing (stochastic) function notation="LaTeX">$f(\cdot)$ , being probability notation="LaTeX">$f(x)$ notation="LaTeX">$x$ defectives tested pool. formulation subsumes variety noiseless and noisy group-testing Our main contributions are follows. Firstly, for any monotone present non-adaptive scheme notation="LaTeX">$1-\varepsilon $ identifies all items. requires most notation="LaTeX">${\mathcal{ O}}\left ({{\Psi (f)} d\log \left ({\frac {n}{\varepsilon }}\right)}\right)$ tests, notation="LaTeX">${\Psi (f)}$ suitably defined “sensitivity parameter” never larger than O}}(d^{1+o(1)})$ but indeed can be substantially smaller . Secondly, argue needs notation="LaTeX">$\Omega ({(1-\varepsilon) {\psi {n} d}\right)}\right)$ ensure high reliability recovery. Here notation="LaTeX">${\psi “concentration (f)}\in \Omega {(1)}$ Thirdly, prove sample-complexity bounds information-theoretically near-optimal sparse-recovery That is, xmlns:xlink="http://www.w3.org/1999/xlink">any “noisy” (i.e., notation="LaTeX">$0 < f(0) f(d) 1$ ), “(one-sided) noiseless” functions either notation="LaTeX">$f(0)=0$ or notation="LaTeX">$f(d)=1$ both) studied literature show notation="LaTeX">$\frac {\Psi \in \Theta (1)$ As by-product tightly characterize heretofore open information-theoretic order-wise well-studied model threshold group-testing. For general (near)-noiseless {\mathcal{ We also demonstrate “natural” test-function whose sample complexity scales “extremally” notation="LaTeX">$\Theta (d^{2}\log n)$ rather (d\log case Some techniques may independent interest – particular achievability delicate saddle-point approximation, impossibility proof relies novel bound relating mutual information pair random variables mean variance specific function, showing upper lower close derive structural results about functions.