Consider the set of source distributions within a fixed maximum relative entropy with respect to a given nominal distribution. Lossless source coding over this relative entropy ball can be approached in more than one way. A problem previously considered is finding a minimax average length source code. The minimizing players are the codeword lengths --- real numbers for arithmetic codes, integers for prefix codes --- while the maximizing players are the uncertain source distributions. Another traditional minimizing objective is the first one considered here, maximum (average) redundancy. This problem reduces to an extension of an exponential Huffman objective treated in the literature but heretofore without direct practical application. In addition to these, this paper examines the related problem of maximal minimax pointwise redundancy and the problem considered by Gawrychowski and Gagie, which, for a sufficiently small relative entropy ball, is equivalent to minimax redundancy. One can consider both Shannon-like coding based on optimal real number ("ideal") codeword lengths and a Huffman-like optimal prefix coding.
[Draft] In this paper, the redundancy of Slepian Wolf coding is revisited. Applying the random binning and converse technique in \cite{yang}, the same results in \cite{he} are obtained with much simpler proofs. Moreover, our results reflect more details about the high-order terms of the coding rate. The redundancy is investigated for both fixed-rate and variable-rate cases. The normal approximation (or dispersion) can also be obtained with minor modification.
This paper deals with the problem of universal lossless coding on a countable infinite alphabet. It focuses on some classes of sources defined by an envelope condition on the marginal distribution, namely exponentially decreasing envelope classes with exponent $α$. The minimax redundancy of exponentially decreasing envelope classes is proved to be equivalent to $\frac{1}{4 α\log e} \log^2 n$. Then a coding strategy is proposed, with a Bayes redundancy equivalent to the maximin redundancy. At last, an adaptive algorithm is provided, whose redundancy is equivalent to the minimax redundancy
The penalty incurred by imposing a finite delay constraint in lossless source coding of a memoryless source is investigated. It is well known that for the so-called block-to-variable and variable-to-variable codes, the redundancy decays at best polynomially with the delay, where in this case the delay is identified with the source block length or maximal source phrase length, respectively. In stark contrast, it is shown that for sequential codes (e.g., a delay-limited arithmetic code) the redundancy can be made to decay exponentially with the delay constraint. The corresponding redundancy-dela…
In this paper we explore tradeoffs, regarding coding performance, between the thickness and spacing of the cutset used in Reduced Cutset Coding (RCC) of a Markov random field image model. Considering MRF models on a square lattice of sites, we show that under a stationarity condition, increasing the thickness of the cutset reduces coding rate for the cutset, increasing the spacing between components of the cutset increases the coding rate of the non-cutset pixels, though the coding rate of the latter is always strictly less than that of the former. We show that the redundancy of RCC can be dec…
We study the effects of finite-precision representation of source's probabilities on the efficiency of classic source coding algorithms, such as Shannon, Gilbert-Moore, or arithmetic codes. In particular, we establish the following simple connection between the redundancy $R$ and the number of bits $W$ necessary for representation of source's probabilities in computer's memory ($R$ is assumed to be small): \begin{equation*} W \lesssim η\log_2 \frac{m}{R}, \end{equation*} where $m$ is the cardinality of the source's alphabet, and $η\leqslant 1$ is an implementation-specific constant. In case of…
This paper presents prefix codes which minimize various criteria constructed as a convex combination of maximum codeword length and average codeword length or maximum redundancy and average redundancy, including a convex combination of the average of an exponential function of the codeword length and the average redundancy. This framework encompasses as a special case several criteria previously investigated in the literature, while relations to universal coding is discussed. The coding algorithm derived is parametric resulting in re-adjusting the initial source probabilities via a weighted pr…
We study an incremental redundancy (IR) cooperative coding scheme for wireless networks. To exploit the spatial diversity benefit we propose a cluster-based collaborating strategy for a quasi-static Rayleigh fading channel model and based on a network geometric distance profile. Our scheme enhances the network performance by embedding an IR cooperative coding scheme into an existing noncooperative route. More precisely, for each hop, we form a collaborating cluster of M-1 nodes between the (hop) sender and the (hop) destination. The transmitted message is encoded using a mother code and partit…
We apply so-called tree straight-line programs to the problem of lossless compression of binary trees. We derive upper bound on the maximal pointwise redundancy (or worst-case redundancy) that improve previous bounds obtained by Zhang, Yang, and Kieffer using directed acyclic graphs. Using this, we obtain universal codes for new classes of structered tree sources.
This paper presents new lower and upper bounds for the compression rate of binary prefix codes optimized over memoryless sources according to various nonlinear codeword length objectives. Like the most well-known redundancy bounds for minimum average redundancy coding - Huffman coding - these are in terms of a form of entropy and/or the probability of an input symbol, often the most probable one. The bounds here, some of which are tight, improve on known bounds of the form L in [H,H+1), where H is some form of entropy in bits (or, in the case of redundancy objectives, 0) and L is the length ob…
Data storage systems are more reliable than their individual components. In order to build highly reliable systems out of less reliable parts, systems introduce redundancy. In replicated systems, objects are simply copied several times with each copy residing on a different physical device. While such an approach is simple and direct, more elaborate approaches such as erasure coding can achieve equivalent levels of data protection while using less redundancy. This report examines the trade-offs in cost and performance between replicated and erasure encoded storage systems.
We consider distributed gradient descent in the presence of stragglers. Recent work on \em gradient coding \em and \em approximate gradient coding \em have shown how to add redundancy in distributed gradient descent to guarantee convergence even if some workers are \em stragglers\em---that is, slow or non-responsive. In this work we propose an approximate gradient coding scheme called \em Stochastic Gradient Coding \em (SGC), which works when the stragglers are random. SGC distributes data points redundantly to workers according to a pair-wise balanced design, and then simply ignores the strag…
In this paper, we investigate the redundancy of universal coding schemes on smooth parametric sources in the finite-length regime. We derive an upper bound on the probability of the event that a sequence of length $n$, chosen using Jeffreys' prior from the family of parametric sources with $d$ unknown parameters, is compressed with a redundancy smaller than $(1-ε)\frac{d}{2}\log n$ for any $ε>0$. Our results also confirm that for large enough $n$ and $d$, the average minimax redundancy provides a good estimate for the redundancy of most sources. Our result may be used to evaluate the performan…
We study universal compression of sequences generated by monotonic distributions. We show that for a monotonic distribution over an alphabet of size $k$, each probability parameter costs essentially $0.5 \log (n/k^3)$ bits, where $n$ is the coded sequence length, as long as $k = o(n^{1/3})$. Otherwise, for $k = O(n)$, the total average sequence redundancy is $O(n^{1/3+ε})$ bits overall. We then show that there exists a sub-class of monotonic distributions over infinite alphabets for which redundancy of $O(n^{1/3+ε})$ bits overall is still achievable. This class contains fast decaying distribut…
Function-Correcting RT Codes: A Coding Framework for Parallel Channels
Central China Normal University（华中师范大学）；LAGA, Department of Mathematics, Universities of Paris VIII and Paris XIII, CNRS, UMR 7539 and Telecom ParisTech（巴黎第八大学和巴黎第十三大学数学系 LAGA，CNRS，UMR 7539 和巴黎电信学院）
As sensing and instrumentation play an increasingly important role in systems controlled over wired and wireless networks, the need to better understand delay-sensitive communication becomes a prime issue. Along these lines, this article studies the operation of data links that employ incremental redundancy as a practical means to protect information from the effects of unreliable channels. Specifically, this work extends a powerful methodology termed sequential differential optimization to choose near-optimal block sizes for hybrid ARQ over erasure channels. In doing so, an interesting connec…
This paper presents new lower and upper bounds for the optimal compression of binary prefix codes in terms of the most probable input symbol, where compression efficiency is determined by the nonlinear codeword length objective of minimizing maximum pointwise redundancy. This objective relates to both universal modeling and Shannon coding, and these bounds are tight throughout the interval. The upper bounds also apply to a related objective, that of dth exponential redundancy.
This document is written in order to establish a common base ground on which the majority of the relevant research about linear fountain codes can be analyzed and compared. As far as I am concerned, there is no unified approach that outlines and compares most of the published linear fountain codes in a single and self-contained framework. This written document has not only resulted in the review of theoretical fundamentals of efficient coding techniques for incremental redundancy and linear fountain coding, but also helped me have a comprehensive reference document and hopefully for many other…
Several studies have identified a significant amount of redundancy in the network traffic. For example, it is demonstrated that there is a great amount of redundancy within the content of a server over time. This redundancy can be leveraged to reduce the network flow by the deployment of memory units in the network. The question that arises is whether or not the deployment of memory can result in a fundamental improvement in the performance of the network. In this paper, we answer this question affirmatively by first establishing the fundamental gains of memory-assisted source compression and …
This paper considers coding for so-called partially stuck (defect) memory cells. Such memory cells can only store partial information as some of their levels cannot be used fully due to, e.g., wearout. First, we present new constructions that are able to mask $u$ partially stuck cells while correcting at the same time $t$ random errors. The process of "masking" determines a word whose entries coincide with writable levels at the (partially) stuck cells. For $u>1$ and alphabet size $q>2$, our new constructions improve upon the required redundancy of known constructions for $t=0$, and require le…
Redundancy is abundant in Fog networks (i.e., many computing and storage points) and grows linearly with network size. We demonstrate the transformational role of coding in Fog computing for leveraging such redundancy to substantially reduce the bandwidth consumption and latency of computing. In particular, we discuss two recently proposed coding concepts, namely Minimum Bandwidth Codes and Minimum Latency Codes, and illustrate their impacts in Fog computing. We also review a unified coding framework that includes the above two coding techniques as special cases, and enables a tradeoff between…
We study the application of polar codes in deletion channels by analyzing the cascade of a binary erasure channel (BEC) and a deletion channel. We show how polar codes can be used effectively on a BEC with a single deletion, and propose a list decoding algorithm with a cyclic redundancy check for this case. The decoding complexity is $O(N^2\log N)$, where $N$ is the blocklength of the code. An important contribution is an optimization of the amount of redundancy added to minimize the overall error probability. Our theoretical results are corroborated by numerical simulations which show that th…
Peer-to-peer distributed storage systems provide reliable access to data through redundancy spread over nodes across the Internet. A key goal is to minimize the amount of bandwidth used to maintain that redundancy. Storing a file using an erasure code, in fragments spread across nodes, promises to require less redundancy and hence less maintenance bandwidth than simple replication to provide the same level of reliability. However, since fragments must be periodically replaced as nodes fail, a key question is how to generate a new fragment in a distributed way while transferring as little data …
Index coding is often studied with the assumption that a single source has all the messages requested by the receivers. We refer to this as \emph{centralized} index coding. In contrast, this paper focuses on \emph{distributed} index coding and addresses the following question: How does the availability of messages at distributed sources (storage nodes) affect the solutions and achievable rates of index coding? An extension to the work of Arbabjolfaei et al. in ISIT 2013 is presented when distributed sources communicate via a semi-deterministic multiple access channel (MAC) to simultaneous rece…
Adaptive coding faces the following problem: given a collection of source classes such that each class in the collection has non-trivial minimax redundancy rate, can we design a single code which is asymptotically minimax over each class in the collection? In particular, adaptive coding makes sense when there is no universal code on the union of classes in the collection. In this paper, we deal with classes of sources over an infinite alphabet, that are characterized by a dominating envelope. We provide asymptotic equivalents for the redundancy of envelope classes enjoying a regular variation …
The order of letters is not always relevant in a communication task. This paper discusses the implications of order irrelevance on source coding, presenting results in several major branches of source coding theory: lossless coding, universal lossless coding, rate-distortion, high-rate quantization, and universal lossy coding. The main conclusions demonstrate that there is a significant rate savings when order is irrelevant. In particular, lossless coding of n letters from a finite alphabet requires Theta(log n) bits and universal lossless coding requires n + o(n) bits for many countable alpha…
As numerous machine learning and other algorithms increase in complexity and data requirements, distributed computing becomes necessary to satisfy the growing computational and storage demands, because it enables parallel execution of smaller tasks that make up a large computing job. However, random fluctuations in task service times lead to straggling tasks with long execution times. Redundancy, in the form of task replication and erasure coding, provides diversity that allows a job to be completed when only a subset of redundant tasks is executed, thus removing the dependency on the straggli…
Motivated from the fact that universal source coding on countably infinite alphabets is not feasible, this work introduces the notion of almost lossless source coding. Analog to the weak variable-length source coding problem studied by Han (IEEE TIT, 2000, 46, 1217-1226), almost lossless source coding aims at relaxing the lossless block-wise assumption to allow an average per-letter distortion that vanishes asymptotically as the block-length tends to infinity. In this setup, we show on one hand that Shannon entropy characterizes the minimum achievable rate (similarly to the case of finite alph…
This paper considers the problem of guessing the realization of a finite alphabet source when some side information is provided. The only knowledge the guesser has about the source and the correlated side information is that the joint source is one among a family. A notion of redundancy is first defined and a new divergence quantity that measures this redundancy is identified. This divergence quantity shares the Pythagorean property with the Kullback-Leibler divergence. Good guessing strategies that minimize the supremum redundancy (over the family) are then identified. The min-sup value measu…
Quantized Indexing is a fast and space-efficient form of enumerative (combinatorial) coding, the strongest among asymptotically optimal universal entropy coding algorithms. The present advance in enumerative coding is similar to that made by arithmetic coding with respect to its unlimited precision predecessor, Elias coding. The arithmetic precision, execution time, table sizes and coding delay are all reduced by a factor O(n) at a redundancy below 2*log(e)/2^g bits/symbol (for n input symbols and g-bit QI precision). Due to its tighter enumeration, QI output redundancy is below that of arithm…
Beijing National Research Center for Information Science and Technology（北京信息科学与技术国家研究中心）；Tsinghua University（清华大学）
Soft compression is a lossless image compression method, which is committed to eliminating coding redundancy and spatial redundancy at the same time by adopting locations and shapes of codebook to encode an image from the perspective of information theory and statistical distribution. In this paper, we propose a new concept, compressible indicator function with regard to image, which gives a threshold about the average number of bits required to represent a location and can be used for revealing the performance of soft compression. We investigate and analyze soft compression for binary image, …
Analog coding decouples the tasks of protecting against erasures and noise. For erasure correction, it creates an "analog redundancy" by means of band-limited discrete Fourier transform (DFT) interpolation, or more generally, by an over-complete expansion based on a frame. We examine the analog coding paradigm for the dual setup of a source with "erasure" side-information (SI) at the encoder. The excess rate of analog coding above the rate-distortion function (RDF) is associated with the energy of the inverse of submatrices of the frame, where each submatrix corresponds to a possible erasure p…
We present a novel lossless universal source coding algorithm that uses parallel computational units to increase the throughput. The length-$N$ input sequence is partitioned into $B$ blocks. Processing each block independently of the other blocks can accelerate the computation by a factor of $B$, but degrades the compression quality. Instead, our approach is to first estimate the minimum description length (MDL) source underlying the entire input, and then encode each of the $B$ blocks in parallel based on the MDL source. With this two-pass approach, the compression loss incurred by using more…
A framework with two scalar parameters is introduced for various problems of finding a prefix code minimizing a coding penalty function. The framework encompasses problems previously proposed by Huffman, Campbell, Nath, and Drmota and Szpankowski, shedding light on the relationships among these problems. In particular, Nath's range of problems can be seen as bridging the minimum average redundancy problem of Huffman with the minimum maximum pointwise redundancy problem of Drmota and Szpankowski. Using this framework, two linear-time Huffman-like algorithms are devised for the minimum maximum p…
Existing gradient coding schemes introduce identical redundancy across the coordinates of gradients and hence cannot fully utilize the computation results from partial stragglers. This motivates the introduction of diverse redundancies across the coordinates of gradients. This paper considers a distributed computation system consisting of one master and $N$ workers characterized by a general partial straggler model and focuses on solving a general large-scale machine learning problem with $L$ model parameters. We show that it is sufficient to provide at most $N$ levels of redundancies for tole…
We consider wiretap channels with uncertainty on the eavesdropper channel under (i) noisy blockwise type II, (ii) compound, or (iii) arbitrarily varying models. We present explicit wiretap codes that can handle these models in a unified manner and only rely on three primitives, namely source coding with side information, universal hashing, and distribution approximation. Our explicit wiretap codes achieve the best known single-letter achievable rates, previously obtained non-constructively, for the models considered. Our results are obtained for strong secrecy, do not require a pre-shared secr…
We study the channel coding problem when errors and uncertainty occur in the encoding process. For simplicity we assume the channel between the encoder and the decoder is perfect. Focusing on linear block codes, we model the encoding uncertainty as erasures on the edges in the factor graph of the encoder generator matrix. We first take a worst-case approach and find the maximum tolerable number of erasures for perfect error correction. Next, we take a probabilistic approach and derive a sufficient condition on the rate of a set of codes, such that decoding error probability vanishes as blockle…
We consider the problem of one-way communication when the recipient does not know exactly the distribution that the messages are drawn from, but has a "prior" distribution that is known to be close to the source distribution, a problem first considered by Juba et al. We consider the question of how much longer the messages need to be in order to cope with the uncertainty about the receiver's prior and the source distribution, respectively, as compared to the standard source coding problem. We consider two variants of this uncertain priors problem: the original setting of Juba et al. in which t…
Independent component analysis (ICA) is a statistical method for transforming an observable multidimensional random vector into components that are as statistically independent as possible from each other.Usually the ICA framework assumes a model according to which the observations are generated (such as a linear transformation with additive noise). ICA over finite fields is a special case of ICA in which both the observations and the independent components are over a finite alphabet. In this work we consider a generalization of this framework in which an observation vector is decomposed to it…
Revisiting Shannon's Source Coding Theorem with Distributional Uncertainty under the Nonlinear Expectation Theory
National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, CAS（中国科学院数学与系统科学研究院交叉科学中心）；School of Information and Communication Engineering, Beijing University of Posts and Telecommunications（北京邮电大学信息与通信工程学院）；University of Chinese Academy of Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Science（中国科学院大学数学与系统科学研究院）
Cut-set bounds on achievable rates for network communication protocols are not in general tight. In this paper we introduce a new technique for proving converses for the problem of transmission of correlated sources in networks, that results in bounds that are tighter than the corresponding cut-set bounds. We also define the concept of "uncertainty region" which might be of independent interest. We provide a full characterization of this region for the case of two correlated random variables. The bounding technique works as follows: on one hand we show that if the communication problem is solv…
An alphabetic binary tree formulation applies to problems in which an outcome needs to be determined via alphabetically ordered search prior to the termination of some window of opportunity. Rather than finding a decision tree minimizing $\sum_{i=1}^n w(i) l(i)$, this variant involves minimizing $\log_a \sum_{i=1}^n w(i) a^{l(i)}$ for a given $a \in (0,1)$. This note introduces a dynamic programming algorithm that finds the optimal solution in polynomial time and space, and shows that methods traditionally used to improve the speed of optimizations in related problems, such as the Hu-Tucker pr…
The effects of quantization and coding on the estimation quality of a Gauss-Markov, namely Ornstein-Uhlenbeck, process are considered. Samples are acquired from the process, quantized, and then encoded for transmission using either infinite incremental redundancy or fixed redundancy coding schemes. A fixed processing time is consumed at the receiver for decoding and sending feedback to the transmitter. Decoded messages are used to construct a minimum mean square error (MMSE) estimate of the process as a function of time. This is shown to be an increasing functional of the age-of-information, d…
Slow running or straggler tasks can significantly reduce computation speed in distributed computation. Recently, coding-theory-inspired approaches have been applied to mitigate the effect of straggling, through embedding redundancy in certain linear computational steps of the optimization algorithm, thus completing the computation without waiting for the stragglers. In this paper, we propose an alternate approach where we embed the redundancy directly in the data itself, and allow the computation to proceed completely oblivious to encoding. We propose several encoding schemes, and demonstrate …
In this paper, we study the problem of lossless universal source coding for stationary memoryless sources on countably infinite alphabets. This task is generally not achievable without restricting the class of sources over which universality is desired. Building on our prior work, we propose natural families of sources characterized by a common dominating envelope. We particularly emphasize the notion of adaptivity, which is the ability to perform as well as an oracle knowing the envelope, without actually knowing it. This is closely related to the notion of hierarchical universal source codin…
We introduce a universal quantization scheme based on random coding, and we analyze its performance. This scheme consists of a source-independent random codebook (typically_mismatched_ to the source distribution), followed by optimal entropy-coding that is_matched_ to the quantized codeword distribution. A single-letter formula is derived for the rate achieved by this scheme at a given distortion, in the limit of large codebook dimension. The rate reduction due to entropy-coding is quantified, and it is shown that it can be arbitrarily large. In the special case of "almost uniform" codebooks (…
The problem of variable length and fixed-distortion universal source coding (or D-semifaithful source coding) for stationary and memoryless sources on countably infinite alphabets ($\infty$-alphabets) is addressed in this paper. The main results of this work offer a set of sufficient conditions (from weaker to stronger) to obtain weak minimax universality, strong minimax universality, and corresponding achievable rates of convergences for the worse-case redundancy for the family of stationary memoryless sources whose densities are dominated by an envelope function (or the envelope family) on $…
