Boolean Compressed Sensing and Noisy Group Testing

The fundamental task of group testing is to recover a small distinguished subset of items from a large population 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. We formulate the group testing problem as a channel coding/decoding problem and derive a single-letter characterization for the total number of tests used to identify the defective set. Although the focus of this paper is primarily on group testing, our main result is generally applicable to other compressive sensing models. The single letter characterization is shown to be order-wise tight for many interesting noisy group testing scenarios. Specifically, we consider an additive Bernoulli($q$) noise model where we show that, for $N$ items and $K$ defectives, the number of tests $T$ is $O(\frac{K\log N}{1-q})$ for arbitrarily small average error probability and $O(\frac{K^2\log N}{1-q})$ for a worst case error criterion. We also consider dilution effects whereby a defective item in a positive pool might get diluted with probability $u$ and potentially missed. In this case, it is shown that $T$ is $O(\frac{K\log N}{(1-u)^2})$ and $O(\frac{K^2\log N}{(1-u)^2})$ for the average and the worst case error criteria, respectively. Furthermore, our bounds allow us to verify existing known bounds for noiseless group testing including the deterministic noise-free case and approximate reconstruction with bounded distortion. Our proof of achievability is based on random coding and the analysis of a Maximum Likelihood Detector, and our information theoretic lower bound is based on Fano's inequality.Comment: In this revision: reorganized the paper, added citations to related work, and fixed some bug

Computationally Tractable Algorithms for Finding a Subset of Non-defective Items from a Large Population

In the classical non-adaptive group testing setup, pools of items are tested together, and the main goal of a recovery algorithm is to identify the "complete defective set" given the outcomes of different group tests. In contrast, the main goal of a "non-defective subset recovery" algorithm is to identify a "subset" of non-defective items given the test outcomes. In this paper, we present a suite of computationally efficient and analytically tractable non-defective subset recovery algorithms. By analyzing the probability of error of the algorithms, we obtain bounds on the number of tests required for non-defective subset recovery with arbitrarily small probability of error. Our analysis accounts for the impact of both the additive noise (false positives) and dilution noise (false negatives). By comparing with the information theoretic lower bounds, we show that the upper bounds on the number of tests are order-wise tight up to a $\log^2K$ factor, where $K$ is the number of defective items. We also provide simulation results that compare the relative performance of the different algorithms and provide further insights into their practical utility. The proposed algorithms significantly outperform the straightforward approaches of testing items one-by-one, and of first identifying the defective set and then choosing the non-defective items from the complement set, in terms of the number of measurements required to ensure a given success rate.Comment: In this revision: Unified some proofs and reorganized the paper, corrected a small mistake in one of the proofs, added more reference

Optimal Nested Test Plan for Combinatorial Quantitative Group Testing

We consider the quantitative group testing problem where the objective is to identify defective items in a given population based on results of tests performed on subsets of the population. Under the quantitative group testing model, the result of each test reveals the number of defective items in the tested group. The minimum number of tests achievable by nested test plans was established by Aigner and Schughart in 1985 within a minimax framework. The optimal nested test plan offering this performance, however, was not obtained. In this work, we establish the optimal nested test plan in closed form. This optimal nested test plan is also order optimal among all test plans as the population size approaches infinity. Using heavy-hitter detection as a case study, we show via simulation examples orders of magnitude improvement of the group testing approach over two prevailing sampling-based approaches in detection accuracy and counter consumption. Other applications include anomaly detection and wideband spectrum sensing in cognitive radio systems

Group Testing with Probabilistic Tests: Theory, Design and Application

Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a classical noiseless group testing setup, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in the sense that the existence of a defective member in a pool results in the test outcome of that pool to be positive. However, this may not be always a valid assumption in some cases of interest. In particular, we consider the case where the defective items in a pool can become independently inactive with a certain probability. Hence, one may obtain a negative test result in a pool despite containing some defective items. As a result, any sampling and reconstruction method should be able to cope with two different types of uncertainty, i.e., the unknown set of defective items and the partially unknown, probabilistic testing procedure. In this work, motivated by the application of detecting infected people in viral epidemics, we design non-adaptive sampling procedures that allow successful identification of the defective items through a set of probabilistic tests. Our design requires only a small number of tests to single out the defective items. In particular, for a population of size $N$ and at most $K$ defective items with activation probability $p$, our results show that $M = O(K^2\log{(N/K)}/p^3)$ tests is sufficient if the sampling procedure should work for all possible sets of defective items, while $M = O(K\log{(N)}/p^3)$ tests is enough to be successful for any single set of defective items. Moreover, we show that the defective members can be recovered using a simple reconstruction algorithm with complexity of $O(MN)$.Comment: Full version of the conference paper "Compressed Sensing with Probabilistic Measurements: A Group Testing Solution" appearing in proceedings of the 47th Annual Allerton Conference on Communication, Control, and Computing, 2009 (arXiv:0909.3508). To appear in IEEE Transactions on Information Theor

On Finding a Subset of Healthy Individuals from a Large Population

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 $N \rightarrow \infty$, to identify $L$ non-defective items out of a population containing $K$ defective items, when the tests are reliable, our results show that $\frac{C_s K}{1-o(1)} (\Phi(\alpha_0, \beta_0) + o(1))$ measurements are sufficient, where $C_s$ is a constant independent of $N, K$ and $L$, and $\Phi(\alpha_0, \beta_0)$ is a bounded function of $\alpha_0 \triangleq \lim_{N\rightarrow \infty} \frac{L}{N-K}$ and $\beta_0 \triangleq \lim_{N\rightarrow \infty} \frac{K} {N-K}$. 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

Estimation of Sparsity via Simple Measurements

We consider several related problems of estimating the 'sparsity' or number of nonzero elements $d$ in a length $n$ vector $\mathbf{x}$ by observing only $\mathbf{b} = M \odot \mathbf{x}$, where $M$ is a predesigned test matrix independent of $\mathbf{x}$, and the operation $\odot$ varies between problems. We aim to provide a $\Delta$-approximation of sparsity for some constant $\Delta$ with a minimal number of measurements (rows of $M$). This framework generalizes multiple problems, such as estimation of sparsity in group testing and compressed sensing. We use techniques from coding theory as well as probabilistic methods to show that $O(D \log D \log n)$ rows are sufficient when the operation $\odot$ is logical OR (i.e., group testing), and nearly this many are necessary, where $D$ is a known upper bound on $d$. When instead the operation $\odot$ is multiplication over $\mathbb{R}$ or a finite field $\mathbb{F}_q$, we show that respectively $\Theta(D)$ and $\Theta(D \log_q \frac{n}{D})$ measurements are necessary and sufficient.Comment: 13 pages; shortened version presented at ISIT 201