Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities

This monograph presents a unified treatment of single- and multi-user problems in Shannon's information theory where we depart from the requirement that the error probability decays asymptotically in the blocklength. Instead, the error probabilities for various problems are bounded above by a non-vanishing constant and the spotlight is shone on achievable coding rates as functions of the growing blocklengths. This represents the study of asymptotic estimates with non-vanishing error probabilities. In Part I, after reviewing the fundamentals of information theory, we discuss Strassen's seminal result for binary hypothesis testing where the type-I error probability is non-vanishing and the rate of decay of the type-II error probability with growing number of independent observations is characterized. In Part II, we use this basic hypothesis testing result to develop second- and sometimes, even third-order asymptotic expansions for point-to-point communication. Finally in Part III, we consider network information theory problems for which the second-order asymptotics are known. These problems include some classes of channels with random state, the multiple-encoder distributed lossless source coding (Slepian-Wolf) problem and special cases of the Gaussian interference and multiple-access channels. Finally, we discuss avenues for further research.Comment: Further comments welcom

Beta-Beta Bounds: Finite-Blocklength Analog of the Golden Formula

It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-blocklength extension of this relation. This extension consists of two elements: 1) a finite-blocklength channel-coding converse bound by Polyanskiy and Verd\'{u} (2014), which involves the ratio of two Neyman-Pearson $\beta$ functions (beta-beta converse bound); and 2) a novel beta-beta channel-coding achievability bound, expressed again as the ratio of two Neyman-Pearson $\beta$ functions. To demonstrate the usefulness of this finite-blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-blocklength extension of Verd\'{u}'s (2002) wideband-slope approximation. The proof parallels the derivation of the latter, with the beta-beta bounds used in place of the golden formula. The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.Comment: to appear in IEEE Transactions on Information Theor

Random Access Channel Coding in the Finite Blocklength Regime

Consider a random access communication scenario over a channel whose operation is defined for any number of possible transmitters. Inspired by the model recently introduced by Polyanskiy for the Multiple Access Channel (MAC) with a fixed, known number of transmitters, we assume that the channel is invariant to permutations on its inputs, and that all active transmitters employ identical encoders. Unlike Polyanskiy, we consider a scenario where neither the transmitters nor the receiver know which transmitters are active. We refer to this agnostic communication setup as the Random Access Channel, or RAC. Scheduled feedback of a finite number of bits is used to synchronize the transmitters. The decoder is tasked with determining from the channel output the number of active transmitters ($k$) and their messages but not which transmitter sent which message. The decoding procedure occurs at a time $n_t$ depending on the decoder's estimate $t$ of the number of active transmitters, $k$, thereby achieving a rate that varies with the number of active transmitters. Single-bit feedback at each time $n_i, i \leq t$, enables all transmitters to determine the end of one coding epoch and the start of the next. The central result of this work demonstrates the achievability on a RAC of performance that is first-order optimal for the MAC in operation during each coding epoch. While prior multiple access schemes for a fixed number of transmitters require $2^k - 1$ simultaneous threshold rules, the proposed scheme uses a single threshold rule and achieves the same dispersion.Comment: Presented at ISIT18', submitted to IEEE Transactions on Information Theor

Fixed-length lossy compression in the finite blocklength regime

This paper studies the minimum achievable source coding rate as a function of blocklength $n$ and probability $\epsilon$ that the distortion exceeds a given level $d$. Tight general achievability and converse bounds are derived that hold at arbitrary fixed blocklength. For stationary memoryless sources with separable distortion, the minimum rate achievable is shown to be closely approximated by $R(d) + \sqrt{\frac{V(d)}{n}} Q^{-1}(\epsilon)$, where $R(d)$ is the rate-distortion function, $V(d)$ is the rate dispersion, a characteristic of the source which measures its stochastic variability, and $Q^{-1}(\epsilon)$ is the inverse of the standard Gaussian complementary cdf

Channel Coding at Low Capacity

Low-capacity scenarios have become increasingly important in the technology of the Internet of Things (IoT) and the next generation of mobile networks. Such scenarios require efficient and reliable transmission of information over channels with an extremely small capacity. Within these constraints, the performance of state-of-the-art coding techniques is far from optimal in terms of either rate or complexity. Moreover, the current non-asymptotic laws of optimal channel coding provide inaccurate predictions for coding in the low-capacity regime. In this paper, we provide the first comprehensive study of channel coding in the low-capacity regime. We will investigate the fundamental non-asymptotic limits for channel coding as well as challenges that must be overcome for efficient code design in low-capacity scenarios.Comment: 39 pages, 5 figure