Boolean compressed sensing: LP relaxation for group testing

We revisit the well-known problem of boolean group testing which attempts to discover a sparse subset of faulty items in a large set of mostly good items using a small number of pooled (or grouped) tests. This problem originated during the second WorldWar, and has been the subject of active research during the 70's, and 80's. Recently, there has been a resurgence of interest due to the striking parallels between group testing and the now highly popular field of compressed sensing. In fact, boolean group testing is nothing but compressed sensing in a different algebra - with boolean `AND' and `OR' operations replacing vector space multiplication and addition. In this paper we review existing solutions for non-adaptive (batch) group testing and propose a linear programming relaxation solution, which has a resemblance to the basis pursuit algorithm for sparse recovery in linear models. We compare its performance to alternative methods for group testing.

[1]  Mikhail Malyutov.  Recovery of sparse active inputs in general systems: A review , 2010, 2010 IEEE Region 8 International Conference on Computational Technologies in Electrical and Electronics Engineering (SIBIRCON).

[2]  Dmitry M. Malioutov,et al.  Belief Propagation and LP Relaxation for Weighted Matching in General Graphs , 2011, IEEE Transactions on Information Theory.

[3]  M. Malyutov.  The separating property of random matrices , 1978 .

[4]  Dino Sejdinovic,et al.  Note on noisy group testing: Asymptotic bounds and belief propagation reconstruction , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[5]  D. Du,et al.  Pooling Designs And Nonadaptive Group Testing: Important Tools For Dna Sequencing , 2006 .

[6]  Venkatesh Saligrama,et al.  Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[7]  Dmitry M. Malioutov,et al.  Sequential Compressed Sensing , 2010, IEEE Journal of Selected Topics in Signal Processing.

[8]  Mikhail Malyutov,et al.  Planning of screening experiments for a nonsymmetric response function , 1980 .

[9]  A. Sterrett.  On the Detection of Defective Members of Large Populations , 1957 .

[10]  Arkadii G. D'yachkov,et al.  A survey of superimposed code theory , 1983 .