Model-Based and Graph-Based Priors for Group Testing

The goal of the group testing problem is to identify a set of defective items within a larger set of items, using suitably-designed tests whose outcomes indicate whether any defective item is present. In this paper, we study how the number of tests can be significantly decreased by leveraging the structural dependencies between the items, i.e., by incorporating prior information. To do so, we pursue two different perspectives: (i) As a generalization of the uniform combinatorial prior, we consider the case that the defective set is uniform over a \emph{subset} of all possible sets of a given size, and study how this impacts the information-theoretic limits on the number of tests for approximate recovery; (ii) As a generalization of the i.i.d.~prior, we introduce a new class of priors based on the Ising model, where the associated graph represents interactions between items. We show that this naturally leads to an Integer Quadratic Program decoder, which can be converted to an Integer Linear Program and/or relaxed to a non-integer variant for improved computational complexity, while maintaining strong empirical recovery performance.Comment: IEEE Transactions on Signal Processin

Group testing:an information theory perspective

The group testing problem concerns discovering a small number of defective items within a large population by performing tests on pools of items. A test is positive if the pool contains at least one defective, and negative if it contains no defectives. This is a sparse inference problem with a combinatorial flavour, with applications in medical testing, biology, telecommunications, information technology, data science, and more. In this monograph, we survey recent developments in the group testing problem from an information-theoretic perspective. We cover several related developments: efficient algorithms with practical storage and computation requirements, achievability bounds for optimal decoding methods, and algorithm-independent converse bounds. We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the {\em rate} of group testing, indicating the amount of information learned per test. Considering both noiseless and noisy settings, we identify several regimes where existing algorithms are provably optimal or near-optimal, as well as regimes where there remains greater potential for improvement. In addition, we survey results concerning a number of variations on the standard group testing problem, including partial recovery criteria, adaptive algorithms with a limited number of stages, constrained test designs, and sublinear-time algorithms.Comment: Survey paper, 140 pages, 19 figures. To be published in Foundations and Trends in Communications and Information Theor

Realizability and uniqueness in graphs

AbstractConsider a finite graph G(V,E). Let us associate to G a finite list P(G) of invariants. To any P the following two natural problems arise: (R) Realizability. Given P, when is P=P(G) for some graph G?, (U) Uniqueness. Suppose P(G)=P(H) for graphs G and H. When does this imply G ≅ H? The best studied questions in this context are the degree realization problem for (R) and the reconstruction conjecture for (U). We discuss the problems (R) and (U) for the degree sequence and the size sequence of induced subgraphs for undirected and directed graphs, concentrating on the complexity of the corresponding decision problems and their connection to a natural search problem on graphs

Algorithms to Exploit Data Sparsity

While data in the real world is very high-dimensional, it generally has some underlying structure; for instance, if we think of an image as a set of pixels with associated color values, most possible settings of color values correspond to something more like random noise than what we typically think of as a picture. With an appropriate transformation of basis, this underlying structure can often be converted into sparsity in data, giving an equivalent representation of the data where the magnitude is large in only a few directions relative to the ambient dimension. This motivates a variety of theoretical questions around designing algorithms that can exploit this data sparsity to achieve better performance than what would be possible naively, and in this thesis we tackle several such questions.We first examine the question of simply approximating the level of sparsity of a signal under several different measurement models, a natural first step if the sparsity is to be exploited by other algorithms. Second, we look at a particular sparse signal recovery problem called nonadaptive probabilistic group testing, and investigate the question of exactly how sparse the signal needs to be before the methods used for recovering sparse signals outperform those used for non-sparse signals. Third, we prove novel upper bounds on the number of measurements needed to recover a sparse signal in the universal one-bit compressed sensing model of sparse signal recovery. Fourth, we give some approximations of an information-theoretic quantity called the index coding rate of a network modeled by a graph, in the special case that the graph is sparse or otherwise highly structured. For each of the problems considered, we also discuss some remaining open questions and conjectures, as well as possible directions towards their solutions

Group Testing in Arbitrary Hypergraphs and Related Combinatorial Structures

We consider a generalization of group testing where the potentially contaminated sets are the members of a given hypergraph ${\cal F}=(V,E)$. This generalization finds application in contexts where contaminations can be conditioned by some kinds of social and geographical clusterings. We study non-adaptive algorithms, two-stage algorithms, and three-stage algorithms. Non-adaptive group testing algorithms are algorithms in which all tests are decided beforehand and therefore can be performed in parallel, whereas two-stage group testing algorithms and three-stage group testing algorithms are algorithms that consist of two stages and three stages, respectively, with each stage being a completely non-adaptive algorithm. In classical group testing, the potentially infected sets are all subsets of up to a certain number of elements of the given input set. For classical group testing, it is known that there exists a correspondence between classical superimposed codes and non-adaptive algorithms, and between two stage algorithms and selectors. Bounds on the number of tests for those algorithms are derived from the bounds on the dimensions of the corresponding combinatorial structures. Obviously, the upper bounds for the classical case apply also to our group testing model. In the present paper, we aim at improving on those upper bounds by leveraging on the characteristics of the particular hypergraph at hand. In order to cope with our version of the problem, we introduce new combinatorial structures that generalize the notions of classical selectors and superimposed codes

ISBIS 2016: Meeting on Statistics in Business and Industry

This Book includes the abstracts of the talks presented at the 2016 International Symposium on Business and Industrial Statistics, held at Barcelona, June 8-10, 2016, hosted at the Universitat Politècnica de Catalunya - Barcelona TECH, by the Department of Statistics and Operations Research. The location of the meeting was at ETSEIB Building (Escola Tecnica Superior d'Enginyeria Industrial) at Avda Diagonal 647. The meeting organizers celebrated the continued success of ISBIS and ENBIS society, and the meeting draw together the international community of statisticians, both academics and industry professionals, who share the goal of making statistics the foundation for decision making in business and related applications. The Scientific Program Committee was constituted by: David Banks, Duke University Amílcar Oliveira, DCeT - Universidade Aberta and CEAUL Teresa A. Oliveira, DCeT - Universidade Aberta and CEAUL Nalini Ravishankar, University of Connecticut Xavier Tort Martorell, Universitat Politécnica de Catalunya, Barcelona TECH Martina Vandebroek, KU Leuven Vincenzo Esposito Vinzi, ESSEC Business Schoo