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Exact Rule Learning via Boolean Compressed Sensing
"... We propose an interpretable rulebased classification system based on ideas from Boolean compressed sensing. We represent the problem of learning individual conjunctive clauses or individual disjunctive clauses as a Boolean group testing problem, and apply a novel linear programming relaxation to fi ..."
Abstract

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We propose an interpretable rulebased classification system based on ideas from Boolean compressed sensing. We represent the problem of learning individual conjunctive clauses or individual disjunctive clauses as a Boolean group testing problem, and apply a novel linear programming relaxation to find solutions. We derive results for exact rule recovery which parallel the conditions for exact recovery of sparse signals in the compressed sensing literature: although the general rule recovery problem is NPhard, under some conditions on the Boolean ‘sensing ’ matrix, the rule can be recovered exactly. This is an exciting development in rule learning where most prior work focused on heuristic solutions. Furthermore we construct rule sets from these learned clauses using set covering and boosting. We show competitive classification accuracy using the proposed approach. 1.
GROTESQUE: Noisy Group Testing (Quick and Efficient)
, 2013
"... Grouptesting refers to the problem of identifying (with high probability) a (small) subset of D defectives from a (large) set of N items via a “small ” number of “pooled ” tests (i.e., tests that have a positive outcome if at least one of the items being tested in the pool is defective, else have a ..."
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Grouptesting refers to the problem of identifying (with high probability) a (small) subset of D defectives from a (large) set of N items via a “small ” number of “pooled ” tests (i.e., tests that have a positive outcome if at least one of the items being tested in the pool is defective, else have a negative outcome). For ease of presentation in this work we focus on the regime when D = O (N1−δ) for some δ> 0. The tests may be noiseless or noisy, and the testing procedure may be adaptive (the pool defining a test may depend on the outcome of a previous test), or nonadaptive (each test is performed independent of the outcome of other tests). A rich body of literature demonstrates that Θ(D log(N)) tests are informationtheoretically necessary and sufficient for the grouptesting problem, and provides algorithms that achieve this performance. However, it is only recently that reconstruction algorithms with computational complexity that is sublinear in N have started being investigated (recent work by [1], [2], [3] gave some of the first such algorithms). In the scenario with adaptive tests with noisy outcomes, we present the first scheme that is simultaneously orderoptimal (up to small constant factors) in both the number of tests and the decoding complexity (O (D log(N)) in both the performance metrics). The total number of stages of our adaptive algorithm is “small ” (O (log(D))). Similarly, in the scenario with nonadaptive tests with noisy outcomes, we present the first scheme that is simultaneously nearoptimal in both the number of tests and the decoding complexity (via an algorithm that requires O (D log(D) log(N)) tests and has a decoding complexity of O(D(logN + log2D)). Finally, we present an adaptive algorithm that only requires 2 stages, and for which both the number of tests and the decoding complexity scale as O(D(logN + log2D)). For all three settings the probability of error of our algorithms scales as O (1/(poly(D)).