Derandomization and Group Testing

The rapid development of derandomization theory, which is a fundamental area in theoretical computer science, has recently led to many surprising applications outside its initial intention. We will review some recent such developments related to combinatorial group testing. In its most basic setting, the aim of group testing is to identify a set of "positive" individuals in a population of items by taking groups of items and asking whether there is a positive in each group. In particular, we will discuss explicit constructions of optimal or nearly-optimal group testing schemes using "randomness-conducting" functions. Among such developments are constructions of error-correcting group testing schemes using randomness extractors and condensers, as well as threshold group testing schemes from lossless condensers.Comment: Invited Paper in Proceedings of 48th Annual Allerton Conference on Communication, Control, and Computing, 201

Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties

We consider the problem of uniform sampling of points on an algebraic variety. Specifically, we develop a randomized algorithm that, given a small set of multivariate polynomials over a sufficiently large finite field, produces a common zero of the polynomials almost uniformly at random. The statistical distance between the output distribution of the algorithm and the uniform distribution on the set of common zeros is polynomially small in the field size, and the running time of the algorithm is polynomial in the description of the polynomials and their degrees provided that the number of the polynomials is a constant

Nearly Optimal Deterministic Algorithm for Sparse Walsh-Hadamard Transform

For every fixed constant $\alpha > 0$, we design an algorithm for computing the $k$-sparse Walsh-Hadamard transform of an $N$-dimensional vector $x \in \mathbb{R}^N$ in time $k^{1+\alpha} (\log N)^{O(1)}$. Specifically, the algorithm is given query access to $x$ and computes a $k$-sparse $\tilde{x} \in \mathbb{R}^N$ satisfying $\|\tilde{x} - \hat{x}\|_1 \leq c \|\hat{x} - H_k(\hat{x})\|_1$, for an absolute constant $c > 0$, where $\hat{x}$ is the transform of $x$ and $H_k(\hat{x})$ is its best $k$-sparse approximation. Our algorithm is fully deterministic and only uses non-adaptive queries to $x$ (i.e., all queries are determined and performed in parallel when the algorithm starts). An important technical tool that we use is a construction of nearly optimal and linear lossless condensers which is a careful instantiation of the GUV condenser (Guruswami, Umans, Vadhan, JACM 2009). Moreover, we design a deterministic and non-adaptive $\ell_1/\ell_1$ compressed sensing scheme based on general lossless condensers that is equipped with a fast reconstruction algorithm running in time $k^{1+\alpha} (\log N)^{O(1)}$ (for the GUV-based condenser) and is of independent interest. Our scheme significantly simplifies and improves an earlier expander-based construction due to Berinde, Gilbert, Indyk, Karloff, Strauss (Allerton 2008). Our methods use linear lossless condensers in a black box fashion; therefore, any future improvement on explicit constructions of such condensers would immediately translate to improved parameters in our framework (potentially leading to $k (\log N)^{O(1)}$ reconstruction time with a reduced exponent in the poly-logarithmic factor, and eliminating the extra parameter $\alpha$). Finally, by allowing the algorithm to use randomness, while still using non-adaptive queries, the running time of the algorithm can be improved to $\tilde{O}(k \log^3 N)$

Applications of Derandomization Theory in Coding

Randomized techniques play a fundamental role in theoretical computer science and discrete mathematics, in particular for the design of efficient algorithms and construction of combinatorial objects. The basic goal in derandomization theory is to eliminate or reduce the need for randomness in such randomized constructions. In this thesis, we explore some applications of the fundamental notions in derandomization theory to problems outside the core of theoretical computer science, and in particular, certain problems related to coding theory. First, we consider the wiretap channel problem which involves a communication system in which an intruder can eavesdrop a limited portion of the transmissions, and construct efficient and information-theoretically optimal communication protocols for this model. Then we consider the combinatorial group testing problem. In this classical problem, one aims to determine a set of defective items within a large population by asking a number of queries, where each query reveals whether a defective item is present within a specified group of items. We use randomness condensers to explicitly construct optimal, or nearly optimal, group testing schemes for a setting where the query outcomes can be highly unreliable, as well as the threshold model where a query returns positive if the number of defectives pass a certain threshold. Finally, we design ensembles of error-correcting codes that achieve the information-theoretic capacity of a large class of communication channels, and then use the obtained ensembles for construction of explicit capacity achieving codes. [This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi

Coding-Theoretic Methods for Sparse Recovery

We review connections between coding-theoretic objects and sparse learning problems. In particular, we show how seemingly different combinatorial objects such as error-correcting codes, combinatorial designs, spherical codes, compressed sensing matrices and group testing designs can be obtained from one another. The reductions enable one to translate upper and lower bounds on the parameters attainable by one object to another. We survey some of the well-known reductions in a unified presentation, and bring some existing gaps to attention. New reductions are also introduced; in particular, we bring up the notion of minimum "L-wise distance" of codes and show that this notion closely captures the combinatorial structure of RIP-2 matrices. Moreover, we show how this weaker variation of the minimum distance is related to combinatorial list-decoding properties of codes.Comment: Added Lemma 34 in the first revision. Original version in Proceedings of the Allerton Conference on Communication, Control and Computing, September 201

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