We consider the nonadaptive group testing with @math items, of which @math are defective. We study a test design in which each item appears in nearly the same number of tests. For each item, we independently pick @math tests uniformly at random with replacement and place the item in those tests. We analyze the performance of these designs with simple and practical decoding algorithms in a range of sparsity regimes and show that the performance is consistently improved in comparison with standard Bernoulli designs. We show that our new design requires roughly 23 fewer tests than a Bernoulli design when paired with the simple decoding algorithms known as combinatorial orthogonal matching pursuit and definite defectives (DD). This gives the best known nonadaptive group testing performance for @math and the best proven performance with a practical decoding algorithm for all @math . We also give a converse result showing that the DD algorithm is optimal with respect to our randomized design when @math . We complement our theoretical results with simulations that show a notable improvement over Bernoulli designs in both sparse and dense regimes.
Kautz and Singleton @cite_3 observed that a construction based on a concatenation of constant-weight codes gives matrices with the so-called @math -disjunctness property (the union of any @math columns does not contain any other column). Such matrices give group testing designs guaranteeing that @math defectives can be recovered with zero probability of error in noiseless group testing (see for example [Chapter 7] du ). However, the group testing designs resulting from the construction of @cite_3 require @math tests. This is an example of the fact that the zero-error criterion requires considerably more tests than the @math required for the error probability tending to zero' criterion (see Definition ) that we study here.
@cite_3
2002912175 2542299727 2962706095 2963583445
We consider the following "efficiently decodable" non-adaptive group testing problem. There is an unknown string x ∈ 0, 1 n with at most d ones in it. We are allowed to test any subset S ⊆ [n] of the indices. The answer to the test tells whether xi = 0 for all i ∈ S or not. The objective is to design as few tests as possible (say, t tests) such that x can be identified as fast as possible (say, poly(t)-time). Efficiently decodable non-adaptive group testing has applications in many areas, including data stream algorithms and data forensics. A non-adaptive group testing strategy can be represented by a t x n matrix, which is the stacking of all the characteristic vectors of the tests. It is well-known that if this matrix is d-disjunct, then any test outcome corresponds uniquely to an unknown input string. Furthermore, we know how to construct d-disjunct matrices with t = O(d2 log n) efficiently. However, these matrices so far only allow for a "decoding" time of O(nt), which can be exponentially larger than poly(t) for relatively small values of d. This paper presents a randomness efficient construction of d-disjunct matrices with t = O(d2 log n) that can be decoded in time poly(d) · t log2 t + O(t2). To the best of our knowledge, this is the first result that achieves an efficient decoding time and matches the best known O(d2 log n) bound on the number of tests. We also derandomize the construction, which results in a polynomial time deterministic construction of such matrices when d = O(log n log log n). A crucial building block in our construction is the notion of (d, l)-list disjunct matrices, which represent the more general "list group testing" problem whose goal is to output less than d + l positions in x, including all the (at most d) positions that have a one in them. List disjunct matrices turn out to be interesting objects in their own right and were also considered independently by [Cheraghchi, FCT 2009]. We present connections between list disjunct matrices, expanders, dispersers and disjunct matrices. List disjunct matrices have applications in constructing (d, l)-sparsity separator structures [Ganguly, ISAAC 2008] and in constructing tolerant testers for Reed-Solomon codes in the data stream model. We investigate recovery of nonnegative vectors from nonadaptive compressive measurements in the presence of noise of unknown power. It is known in the literature that under additional assumptions on the measurement design recovery of such vectors is possible with nonnegative least squares without any regularization. We show that uniqueness results known for the noiseless case carry over to robust guarantees in the noisy setting. We present guarantees which hold instantaneously by connecting the relation to the robust nullspace property. As an important example, we prove that an m × n random iid. 0 1-valued Bernoulli matrix with m = O(s log(n)) rows admits the robust nullspace property with high probability and meets the design requirements for nonnegative least squares recovery. Our analysis is motivated by applications in wireless network activity detection. The group testing problem is to identify a population of K defective items from a set of n items by pooling groups of items. The result of a test for a group of items is positive if any of the items in the group is defective and negative otherwise. The goal is to judiciously group subsets of items such that defective items can be reliably recovered using the minimum number of tests, while also having a low-complexity decoder. We describe SAFFRON (Sparse-grAph codes Framework For gROup testiNg), a non-adaptive group testing scheme that recovers at least a (1 − ϵ)-fraction (for any arbitrarily small ϵ > 0) of K defective items with high probability with m = 6C(ϵ)K log 2 n tests, where C(ϵ) is a precisely characterized constant that depends only on o. For instance, it can provably recover at least (1 − 10−6)K defective items with m ≃ 68K log 2 n tests. The computational complexity of the decoding algorithm is O(K log n), which is order-optimal. Further, we describe a systematic methodology to robustify SAFFRON such that it can reliably recover the set of K defective items even in the presence of erroneous or noisy test results. We also propose Singleton-Only-SAFFRON, a variant of SAFFRON, that recovers all the K defective items with m = 2e(1+α)K log K log 2 n tests with probability 1 − O(1 Kα), where α > 0 is a constant. Our key intellectual contribution involves the pioneering use of powerful density-evolution methods of modern coding theory (e.g. sparse-graph codes) for efficient group testing design and performance analysis. Abstract We study the recovery of Hermitian low rank matrices X ∈ C n × n from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form a j a j ⁎ for some measurement vectors a 1 , … , a m , i.e., the measurements are given by b j = tr ( X a j a j ⁎ ) . The case where the matrix X = x x ⁎ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements b j = | 〈 x , a j 〉 | 2 ) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors a j , j = 1 , … , m , being chosen independently at random according to a standard Gaussian distribution, or a j being sampled independently from an (approximate) complex projective t-design with t = 4 . In the Gaussian case, we require m ≥ C r n measurements, while in the case of 4-designs we need m ≥ Cr n log ⁡ ( n ) . Our results are uniform in the sense that one random choice of the measurement vectors a j guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Krahmer and Kueng. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.