Non-adaptive group testing: Explicit bounds and novel algorithms

Abstract

1 . We present computationally efficient and provably correct algorithms with near-optimal sample-complexity for noisy non-adaptive group testing. Group testing involves grouping arbitrary subsets of items into pools. Each pool is then tested to identify the defective items, which are usually assumed to be sparsely distributed. We consider random non-adaptive pooling where pools are selected randomly and independently of the test outcomes. Our noisy scenario accounts for both false negatives and false positives for the test outcomes. Inspired by compressive sensing algorithms we introduce four novel computationally efficient decoding algorithms for group testing, CBP via Linear Programming (CBP-LP), NCBP-LP (Noisy CBP-LP), and the two related algorithms NCBP-SLP+ and NCBP-SLP-("Simple" NCBP-LP). The first of these algorithms deals with the noiseless measurement scenario, and the next three with the noisy measurement scenario. We derive explicit sample-complexity bounds-with all constants made explicit-for these algorithms as a function of the desired error probability; the noise parameters; the number of items; and the size of the defective set (or an upper bound on it). We show that the samplecomplexities of our algorithms are near-optimal with respect to known information-theoretic bounds.

Semiquantitative Group Testing

“…On the other hand, decoders for CGT were extensively investigated in the literature (e.g. [50]- [53]). Although these algorithms perform well for CGT schemes, due to the more complicated nature of SQGT, their direct application to SQGT does not appear to be plausible.…”

Active hypothesis testing on a tree: Anomaly detection under hierarchical observations

“…Most work on Boolean group testing assumes errorfree test outcomes. There are several recent studies on noisy group testing that assume the presence of one-sided noise [15], [16] or the symmetric case with equal size-independent false alarm and miss detection probabilities [17], [18]. In some extened group testing models such as the noisy quantitative group testing [19] and threshold group testing [20], the issue of sample complexity in terms of detection accuracy is absent in the basic formulation.…”

Non-Adaptive Group Testing: Explicit Bounds and Novel Algorithms

“…In contrast, in group testing, the measurements are non-linear and boolean. Nonetheless, our contribution here is to show 2 We wish to highlight the difference between noise and errors. We use the former term to refer to noise in the outcomes of the group-test, regardless of the group-testing algorithm used.…”

“…1 Since the measurements are noisy, the problem of estimating the set of defective items is more challenging. 2 In this work we focus primarily on noise models wherein a positive test outcome is a false positive with the same probability as a negative test outcome being a noisy positive. We do this primarily for ease of analysis (as, for instance, the Binary Symmetric Channel noise model is much more extensively studied in the literature than other less symmetric models), though our techniques also work for asymmetric error probabilities, and for activation noise.…”

“…In Section III, we describe the main results of this 5 In the noiseless setting prior work by [23], [24] had considered a similar LP approach (though no detailed analysis was presented). For the scenario with noisy measurements, in parallel with the conference version of our work presented at [2], a similar algorithm was considered in [25] -however, the proof sketch presented therein requires disjunct measurement matrices, which are known not to be order-optimal in sample-complexity in the "small-error" case [9], [10]. 6 Prior work, for instance by [26], had also considered group testing that was similarly universal in D. The T × n binary group-testing matrix represents the items being tested in each test, the length-n binary input vector x is a weight d vector encoding the locations of the d defective items in D, the length-T binary vector y denotes the outcomes of the group tests in the absence of noise, the length-T binary noisy result vectorŷ denotes the actually observed noisy outcomes of the group tests, as the result of the noiseless result vector being perturbed by the length-T binary noise vector ν.…”