There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Non-orthogonal multiple access (NOMA) is one of the promising radio access techniques for performance enhancement in next-generation cellular communications. Compared to orthogonal frequency division multiple access, which is a well-known high-capacity orthogonal multiple access technique, NOMA offers a set of desirable benefits, including greater spectrum efficiency. There are different types of NOMA techniques, including power-domain and code-domain. This paper primarily focuses on power-domain NOMA that utilizes superposition coding at the transmitter and successive interference cancellation at the receiver. Various researchers have demonstrated that NOMA can be used effectively to meet both network-level and user-experienced data rate requirements of fifth-generation (5G) technologies. From that perspective, this paper comprehensively surveys the recent progress of NOMA in 5G systems, reviewing the state-of-the-art capacity analysis, power allocation strategies, user fairness, and user-pairing schemes in NOMA. In addition, this paper discusses how NOMA performs when it is integrated with various proven wireless communications techniques, such as cooperative communications, multiple-input multiple-output, beamforming, space-time coding, and network coding among others. Furthermore, this paper discusses several important issues on NOMA implementation and provides some avenues for future research.

Power-Domain Non-Orthogonal Multiple Access (NOMA) in 5G Systems: Potentials and Challenges

The CDMA channel with randomly and independently chosen spreading sequences accurately models the situation where pseudonoise sequences span many symbol periods. Furthermore, its analysis provides a comparison baseline for CDMA channels with deterministic signature waveforms spanning one symbol period. We analyze the spectral efficiency (total capacity per chip) as a function of the number of users, spreading gain, and signal-to-noise ratio, and we quantify the loss in efficiency relative to an optimally chosen set of signature sequences and relative to multiaccess with no spreading. White Gaussian background noise and equal-power synchronous users are assumed. The following receivers are analyzed: (a) optimal joint processing, (b) single-user matched filtering, (c) decorrelation, and (d) MMSE linear processing.

/pdf/spectral-efficiency-of-cdma-with-random-spreading-4btfeuvqxo.pdf

Spectral efficiency of CDMA with random spreading

Non-orthogonal multiple access (NOMA) is one of the promising radio access techniques for performance enhancement in next-generation cellular communications. Compared to orthogonal frequency division multiple access (OFDMA), which is a well-known high-capacity orthogonal multiple access (OMA) technique, NOMA offers a set of desirable benefits, including greater spectrum efficiency. There are different types of NOMA techniques, including power-domain and code-domain. This paper primarily focuses on power-domain NOMA that utilizes superposition coding (SC) at the transmitter and successive interference cancellation (SIC) at the receiver. Various researchers have demonstrated that NOMA can be used effectively to meet both network-level and user-experienced data rate requirements of fifth-generation (5G) technologies. From that perspective, this paper comprehensively surveys the recent progress of NOMA in 5G systems, reviewing the state-of-the-art capacity analysis, power allocation strategies, user fairness, and user-pairing schemes in NOMA. In addition, this paper discusses how NOMA performs when it is integrated with various proven wireless communications techniques, such as cooperative communications, multiple input multiple output (MIMO), beamforming, space time coding, and network coding, among others. Furthermore, this paper discusses several important issues on NOMA implementation and provides some avenues for future research.

This book provides a comprehensive description of a new method of proving the central limit theorem, through the use of apparently unrelated results from information theory. It gives a basic introduction to the concepts of entropy and Fisher information, and collects together standard results concerning their behaviour. It brings together results from a number of research papers as well as unpublished material, showing how the techniques can give a unified view of limit theorems.

Information Theory And The Central Limit Theorem

We consider the problem of nonadaptive noiseless group testing of $N$ items of which $K$ are defective. We describe four detection algorithms, the ${\tt COMP}$ algorithm of Chan et al. , two new algorithms, ${\tt DD}$ and ${\tt SCOMP}$ , which require stronger evidence to declare an item defective, and an essentially optimal but computationally difficult algorithm called ${\tt SSS}$ . We consider an important class of designs for the group testing problem, namely those in which the test structure is given via a Bernoulli random process. In this class of Bernoulli designs, by considering the asymptotic rate of these algorithms, we show that ${\tt DD}$ outperforms ${\tt COMP}$ , that ${\tt DD}$ is essentially optimal in regimes where $K\geq\sqrt N$ , and that no algorithm can perform as well as the best nonrandom adaptive algorithms when $K>N^{0.35}$ . In simulations, we see that ${\tt DD}$ and ${\tt SCOMP}$ far outperform ${\tt COMP}$ , with ${\tt SCOMP}$ very close to the optimal ${\tt SSS}$ , especially in cases with larger $K$ .

/pdf/group-testing-algorithms-bounds-and-simulations-23n8aykym3.pdf

Group Testing Algorithms: Bounds and Simulations

We give conditions for an O(1/n) rate of convergence of Fisher information and relative entropy in the Central Limit Theorem. We use the theory of projections in L
 2 spaces and Poincare inequalities, to provide a better understanding of the decrease in Fisher information implied by results of Barron and Brown. We show that if the standardized Fisher information ever becomes finite then it converges to zero.

Fisher Information inequalities and the Central Limit Theorem

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.

/pdf/group-testing-an-information-theory-perspective-4bqcwcbxrv.pdf

Group Testing: An Information Theory Perspective

/pdf/group-testing-an-information-theory-perspective-21ckdmzbna.pdf

Oliver Johnson

Papers

Information Theory And The Central Limit Theorem

Group Testing Algorithms: Bounds and Simulations

Fisher Information inequalities and the Central Limit Theorem

Group Testing: An Information Theory Perspective

Group Testing: An Information Theory Perspective