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This paper studies a decentralized wireless network of asynchronous transmitter–receiver pairs with burst transmissions. Each receiver learns about the number of active users, channel coefficients, and mutual delays based on locally available measurements. The estimates for the mutual delays are not perfect, however, they are reliable enough to guarantee successful decoding. Two signalling schemes are addressed, namely, randomized masking (RM) and reduced cycle transmission (RCT). Under RM, the $n$ symbols of a codeword are generated according to a Bernoulli–Gaussian distribution with activity factor $0<\theta \leq 1$ . This is in contrast to RCT where each codeword consists of $\lfloor \theta n\rfloor $ Gaussian symbols followed by $n-\lfloor \theta n\rfloor $ zeros. Assuming the transmitters are unaware of the number of users, channel coefficients, and mutual delays, the probability of outage under RM is considerably lower compared with RCT if the signal-to-noise ratio (SNR) is sufficiently large. A generalized RCT scheme is also examined where the $n-\lfloor \theta n\rfloor $ zero symbols are not necessarily located at the end of a codeword. In the asymptote of large SNR, the outage probability becomes vanishingly small under RM, however, it is bounded away from zero for generalized RCT regardless of the value of SNR. [ABSTRACT FROM PUBLISHER]

We consider the basic bidirectional relaying problem, in which two users in a wireless network wish to exchange messages through an intermediate relay node. In the compute-and-forward strategy, the relay computes a function of the two messages using the naturally occurring sum of symbols simultaneously transmitted by user nodes in a Gaussian multiple-access channel (MAC), and the computed function value is forwarded to the user nodes in an ensuing broadcast phase. In this paper, we study the problem under an additional security constraint, which requires that each user’s message be kept secure from the relay. We consider two types of security constraints: 1) perfect secrecy, in which the MAC channel output seen by the relay is independent of each user’s message and 2) strong secrecy, which is a form of asymptotic independence. We propose a coding scheme based on nested lattices, the main feature of which is that given a pair of nested lattices that satisfy certain goodness properties, we can explicitly specify probability distributions for randomization at the encoders to achieve the desired security criteria. In particular, our coding scheme guarantees perfect or strong secrecy even in the absence of channel noise. The noise in the channel only affects reliability of computation at the relay, and for Gaussian noise, we derive achievable rates for reliable and secure computation. We also present an application of our methods to the multihop line network in which a source needs to transmit messages to a destination through a series of intermediate relays. [ABSTRACT FROM AUTHOR]

Guyon (Guyon J. Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann Appl Probab, 2007, 17: 1538-1569) introduced an important model for homogeneous bifurcating Markov chains indexed by a binary tree taking values in general state space and studied their limit theorems. The results were applied to detect cellular aging. In this paper, we define a discrete form of nonhomogeneous bifurcating Markov chains indexed by a binary tree and discuss the equivalent properties for them. The strong law of large numbers and the entropy ergodic theorem are studied for these Markov chains with finite state space. In contrast to previous work, we use a new approach to prove the main results of this paper. [ABSTRACT FROM AUTHOR]

This paper studies sequential methods for recovery of sparse signals in high dimensions. When compared with fixed sample size procedures, in the sparse setting, sequential methods can result in a large reduction in the number of samples needed for reliable signal support recovery. Starting with a lower bound, we show any coordinate-wise sequential sampling procedure fails in the high dimensional limit provided the average number of measurements per dimension is less then \log (s)/D(P0||P1) , where s is the level of sparsity and D(P_{0}||P_{1}) is the Kullback–Leibler divergence between the underlying distributions. A series of sequential probability ratio tests, which require complete knowledge of the underlying distributions is shown to achieve this bound. Motivated by real-world experiments and recent work in adaptive sensing, we introduce a simple procedure termed sequential thresholding, which can be implemented when the underlying testing problem satisfies a monotone likelihood ratio assumption. Sequential thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than \log (s)/D(P0||P1) , achieving the lower bound. For comparison, we show any nonsequential procedure fails provided the number of measurements grows at a rate less than \log (n)/D(P_{1}||P_{0})$ , where $n$ is the total dimension of the problem. [ABSTRACT FROM AUTHOR]

We consider a cooperative two-user multiaccess channel in which the transmission is controlled by a random state. Both encoders transmit a common message and, one of the encoders also transmits an individual message. We study the capacity region of this communication model for different degrees of availability of the states at the encoders, causally or strictly causally. In the case in which the states are revealed causally to both encoders but not to the decoder we find an explicit characterization of the capacity region in the discrete memoryless case. In the case in which the states are revealed only strictly causally to both encoders, we establish inner and outer bounds on the capacity region. The outer bound is nontrivial, and has a relatively simple form. It has the advantage of incorporating only one auxiliary random variable. In particular, it suggests that there is none, or at best only little, to gain from having the encoder that transmits both messages also sending an individual description of the state to the receiver, in addition to the compressed version that is sent cooperatively with the other encoder. We then introduce a class of cooperative multiaccess channels with states known strictly causally at both encoders for which the inner and outer bounds agree, and so we characterize the capacity region for this class. In this class of channels, the state can be obtained as a deterministic function of the channel inputs and output. We also study the model in which the states are revealed, strictly causally, in an asymmetric manner, to only one encoder. Throughout this paper, we discuss a number of examples, and compute the capacity region of some of these examples. The results shed more light on the utility of delayed channel state information for increasing the capacity region of state-dependent cooperative multiaccess channels, and tie with recent progress in this framework. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the structure of infeasibility results in network information theory, based on a notion of information state. We also discuss ideas for generalizing a known outer bound for lossless transmission of independent sources over a network to one of lossy transmission of dependent sources over the same network. To concretely demonstrate this, we apply our ideas and prove new results for lossy transmission of dependent sources by generalizing: 1) the cut-set bound; 2) the best known outer bound on the capacity region of a general broadcast channel; and 3) the outer bound part of the result of Maric, Yates, and Kramer on strong interference channels with a common message. [ABSTRACT FROM AUTHOR]

This paper investigates the downlink of a cloud radio access network (C-RAN) in which a central processor communicates with two mobile users through two base stations (BSs). The BSs act as relay nodes and cooperate with each other through error-free rate-limited links. We develop and analyze two coding schemes for this scenario. The first coding scheme modifies the Liu–Kang scheme (to make it amenable to a rigorous analysis) and extends it to introduce common codewords and to apply for downlink C-RAN with BS-to-BS cooperation. This first coding scheme enables arbitrary correlation among the auxiliary codewords that are recovered by the BSs. We show that this scheme improves over previous schemes for various instances of Gaussian C-RAN channels. In particular, in many scenarios, the scheme can better exploit the possibility of BS-to-BS cooperation than other schemes. The second coding scheme extends the distributed decode and forward (DDF) scheme by means of Gray–Wyner compression and by exploiting the cooperation links between BSs. In addition and as a separate extension, we provide an improved capacity approximation for the DDF strategy for the capacity of a general $N$ -BS $L$ -user C-RAN model in the memoryless Gaussian case. [ABSTRACT FROM AUTHOR]

High-order cumulants and high-order spectra play an important role in the theory of stationary processes. For nonlinear processes, it is quite challenging to develop an asymptotic theory for their estimators. This paper presents a systematic asymptotic theory for estimators of high-order moments for a general class of stationary processes, using the framework of functional dependence measures. In particular, we prove the asymptotic normality of the estimators and establish a uniform convergence rate. We also provide a sufficient condition for the summability of high-order cumulants. Based on the latter, we prove consistency and asymptotic normality of the third-order spectra or bispectrum estimators under mild and easily verifiable conditions. [ABSTRACT FROM AUTHOR]

Rivest and Shamir introduced a write-once memory (WOM), a model of storage devices whose storage elements have restrictions on state transitions, and they presented some coding methods to reuse a WOM. An interesting question about a WOM is how efficiently we can reuse it with the best coding method, and as an answer to the question, Fu and Han Vinck determined the capacity of Fiat and Shamir’s generalized WOMs. In this paper, we extend their results, introducing write-constrained memories (WCMs) that consider state transition costs, and determining the capacity of WCMs under a certain type of cost constraints. [ABSTRACT FROM AUTHOR]

This paper addresses the correspondence between linear inequalities for Shannon entropy and differential entropy for sums of independent group-valued random variables. We show that any balanced (with the sum of coefficients being zero) linear inequality for Shannon entropy holds if and only if its differential entropy counterpart also holds; moreover, any linear inequality for differential entropy must be balanced. In particular, our result shows that recently proved differential entropy inequalities by Kontoyiannis and Madiman can be deduced from their discrete counterparts due to Tao in a unified manner. Generalizations to certain abelian groups are also obtained. Our proof of extending inequalities for Shannon entropy to differential entropy relies on a result of Rényi which relates the Shannon entropy of a finely discretized random variable to its differential entropy and also helps in establishing that the entropy of the sum of quantized random variables is asymptotically equal to that of the quantized sum; the converse uses the asymptotics of the differential entropy of convolutions with weak additive noise. [ABSTRACT FROM PUBLISHER]

A measure of correlation is said to have the tensorization property if it does not change when computed for i.i.d. copies. More precisely, a measure of correlation between two random variables X, Y denoted by \rho (X, Y) , has the tensorization property if \rho (X^n, Y^n)=\rho (X, Y) where (X^n, Y^n) denotes $n$ i.i.d. copies of (X, Y)$ . Two well-known examples of such measures are the maximal correlation and the hypercontractivity ribbon (HC ribbon). We show that the maximal correlation and the HC ribbon are special cases of the new notion of \Phi $ -ribbons, defined in this paper for a class of convex functions \Phi $ . \Phi $ -ribbon reduces to the HC ribbon and the maximal correlation for special choices of \Phi $ , and is a measure of correlation with the tensorization property. We show that the \Phi $ -ribbon also characterizes the recently introduced and introduce an equivalent characterization of this ribbon. [ABSTRACT FROM AUTHOR]

It is well known that the entropy $H(X)$ of a discrete random variable $X$ is always greater than or equal to the entropy $H(f(X))$ of a function $f$ of $X$ , with equality if and only if $f$ is one-to-one. In this paper, we give tight bounds on $H(f(X))$ , when the function $f$ is not one-to-one, and we illustrate a few scenarios, where this matters. As an intermediate step toward our main result, we derive a lower bound on the entropy of a probability distribution, when only a bound on the ratio between the maximal and minimal probabilities is known. The lower bound improves on previous results in the literature, and it could find applications outside the present scenario. [ABSTRACT FROM AUTHOR]

We consider the problem of mean-square estimation of the state of a discrete time dynamical system having additive non-Gaussian noise. We assume that the noise has the structure that at each time instant, it is a projection of a fixed high-dimensional noise vector with a log-concave density. We derive conditions which guarantee that, as the dimension of this noise vector grows large, the optimal estimator of the problem with Gaussian noise becomes near-optimal for the problem with non-Gaussian noise. The results are derived by first showing an asymptotic Gaussian lower bound on the minimum error which holds even for nonlinear systems. The bound is asymptotically tight for linear systems. For nonlinear systems, this bound is tight provided and the noise has a strongly log-concave density with some additional structure. These results imply that the estimate obtained by employing the Gaussian estimator on the non-Gaussian problem satisfies an approximate orthogonality principle, and the difference between this estimate and the optimal estimate vanishes strongly in L^2 . For linear systems with high-dimensional log-concave noise of this structure, we get that the Kalman filter serves as a near-optimal estimator. A key ingredient in the proofs is a recent central limit theorem of Eldan and Klartag. The minimum mean-square error is in general neither weakly upper semicontinuous nor weakly lower semicontinuous. Our results show that in the setting we consider, it is weakly continuous.

We then consider mean-square estimation over a noisy network, where the problem is to jointly design strategies for sensing and estimation to minimize a composite quadratic cost. These problems do not have a static information structure, whereby the previously derived results do not apply. We consider two different cases—when the estimation part of the problem corresponds to estimating the original source sensed by the sensor, and when this part corresponds to the estimation of the action of the sensor. In the former case, we show that results analogous to the above continue to hold. The latter case corresponds to a variant of Witsenhausen’s problem, wherein we show only partial results. [ABSTRACT FROM AUTHOR]

Usually, it is very difficult to determine the exact distribution for a test statistic. In this paper, asymptotic distributions of locally most powerful invariant test for independence of complex Gaussian vectors are developed. In particular, its cumulative distribution function (CDF) under the null hypothesis is approximated by a function of chi-squared CDFs. Moreover, the CDF corresponding to the non-null distribution is expressed in terms of non-central chi-squared CDFs for close hypothesis, and Gaussian CDF as well as its derivatives for far hypothesis. The results turn out to be very accurate in terms of fitting their empirical counterparts. Closed-form expression for the detection threshold is also provided. Numerical results are presented to validate our theoretical findings. [ABSTRACT FROM PUBLISHER]

We revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis. We introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms. In this case, we find the threshold rate, the optimal error and strong converse exponents (at large deviations from the threshold), and the second order asymptotics (at small deviations from the threshold). We apply our results to find the operational interpretations of various Rényi information measures. In case the alternative hypothesis is comprised of bipartite product distributions, we find that the optimal error and strong converse exponents are determined by the variations of Rényi mutual information. In case the alternative hypothesis consists of tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by the variations of Rényi conditional mutual information. In either case, the relevant notion of Rényi mutual information depends on the precise choice of the alternative hypothesis. As such, this paper also strengthens the view that different definitions of Rényi mutual information, conditional entropy, and conditional mutual information are adequate depending on the context in which the measures are used. [ABSTRACT FROM AUTHOR]

In this paper, we propose an open-loop unequal-error-protection querying policy based on superposition coding for the noisy 20 questions problem. In this problem, a player wishes to successively refine an estimate of the value of a continuous random variable by posing binary queries and receiving noisy responses. When the queries are designed non-adaptively as a single block and the noisy responses are modeled as the output of a binary symmetric channel, the 20 questions problem can be mapped to an equivalent problem of channel coding with unequal error protection (UEP). A new non-adaptive querying strategy based on UEP superposition coding is introduced, whose estimation error decreases with an exponential rate of convergence that is significantly better than that of the UEP repetition coding introduced by Variani et al. (2015). With the proposed querying strategy, the rate of exponential decrease in the number of queries matches the rate of a closed-loop adaptive scheme, where queries are sequentially designed with the benefit of feedback. Furthermore, the achievable error exponent is significantly better than that of random block codes employing equal error protection. [ABSTRACT FROM AUTHOR]

The entropy of a random variable is well-known to equal the exponential growth rate of the volumes of its typical sets. In this paper, we show that for any log-concave random variable X , the sequence of the \lfloor n\theta \rfloor ^{\text {th}} intrinsic volumes of the typical sets of X in dimensions n \geq 1 grows exponentially with a well-defined rate. We denote this rate by h_{X}(\theta ) , and call it the \theta ^{\text {th}} intrinsic entropy of X . We show that h_{X}(\theta ) is a continuous function of \theta $ over the range [0, 1], thereby providing a smooth interpolation between the values 0 and $h(X)$ at the endpoints 0 and 1, respectively. [ABSTRACT FROM PUBLISHER]

Let X and Y be dependent random variables. This paper considers the problem of designing a scalar quantizer for Y to maximize the mutual information between the quantizer’s output and X, and develops fundamental properties and bounds for this form of quantization, which is connected to the log-loss distortion criterion. The main focus is the regime of low ${I}({X};{Y})$ , where it is shown that, if X is binary, a constant fraction of the mutual information can always be preserved using $\mathcal {O}(\log (1/{I}({X};{Y})))$ quantization levels, and there exist distributions for which this many quantization levels are necessary. Furthermore, for larger finite alphabets $2 < |\mathcal {X}| < \infty $ , it is established that an $\eta $ -fraction of the mutual information can be preserved using roughly $(\log (| \mathcal {X} | /{I}({X};{Y})))^{\eta \cdot (|\mathcal {X}| - 1)}$ quantization levels. [ABSTRACT FROM AUTHOR]

This paper provides a precise error analysis for the maximum likelihood estimate $\hat {\textit a}_{\text {ML}}({\textit u}_{1}^{\textit n})$ of the parameter ${\textit a}$ given samples ${\textit u}_{1}^{\textit n} = ({\textit u}_{1}, \ldots, {\textit u}_{\textit n})'$ drawn from a nonstationary Gauss-Markov process ${\textit U}_{\textit i} = {\textit a} {\textit U}_{\textit i-1} + {\textit Z}_{\textit i},\,\,{\textit i}\geq 1$ , where ${\textit U}_{0} = 0$ , ${\textit a}> 1$ , and ${\textit Z}_{\textit i}$ ’s are independent Gaussian random variables with zero mean and variance $\sigma ^{2}$. We show a tight nonasymptotic exponentially decaying bound on the tail probability of the estimation error. Unlike previous works, our bound is tight already for a sample size of the order of hundreds. We apply the new estimation bound to find the dispersion for lossy compression of nonstationary Gauss-Markov sources. We show that the dispersion is given by the same integral formula that we derived previously for the asymptotically stationary Gauss-Markov sources, i.e., $|{\textit a}| < 1$. New ideas in the nonstationary case include separately bounding the maximum eigenvalue (which scales exponentially) and the other eigenvalues (which are bounded by constants that depend only on a) of the covariance matrix of the source sequence, and new techniques in the derivation of our estimation error bound. [ABSTRACT FROM AUTHOR]

In many information-theoretic channel coding problems, adding an input cost constraint to the operational setup amounts to restricting the optimization domain in the capacity formula. This paper shows that, in contrast to common belief, such a simple modification does not hold for the cost-constrained (CC) wiretap channel (WTC). The secrecy-capacity of the discrete memoryless (DM) WTC without cost constraints is described by a single auxiliary random variable. For the CC DM-WTC, however, we show that two auxiliaries are necessary to achieve capacity. Specifically, we first derive the secrecy-capacity formula, proving the direct part via superposition coding. Then, we provide an example of a CC DM-WTC whose secrecy-capacity cannot be achieved using a single auxiliary. This establishes the fundamental role of superposition coding over CC WTCs. [ABSTRACT FROM AUTHOR]

In group testing, simple binary-output tests are designed to identify a small number $t$ of defective items that are present in a large population of $N$ items. Each test takes as input a group of items and produces a binary output indicating whether the group is free of the defective items or contains one or more of them. In this paper, we study a relaxation of the combinatorial group testing problem. A matrix is called $(t,\epsilon )$ -disjunct if it gives rise to a nonadaptive group testing scheme with the property of identifying a uniformly random $t$ -set of defective subjects out of a population of size $N$ with false positive probability of an item at most $\epsilon $ . We establish a new connection between $(t,\epsilon )$ -disjunct matrices and error correcting codes based on the dual distance of the codes and derive estimates of the parameters of codes that give rise to such schemes. Our methods rely on the moments of the distance distribution of codes and inequalities for moments of sums of independent random variables. We also provide a new connection between group testing schemes and combinatorial designs. [ABSTRACT FROM PUBLISHER]

The problem of characterizing the fundamental tradeoff between storage and repair bandwidth of exact-repair regenerating codes against a passive eavesdropper is studied. The eavesdropper is assumed to be capable of observing the data stored in a fixed number of nodes and the data involved in the repair of these nodes. In this paper, the tradeoff for regenerating codes with small parameters is characterized, and then, the results are extended to some general settings. [ABSTRACT FROM PUBLISHER]

In this paper, we consider a cache aided network in which each user is assumed to have individual caches, while upon users’ requests, an update message is sent through a common link to all users. First, we formulate a general information theoretic setting that represents the database as a discrete memoryless source, and the users’ requests as side information that is available everywhere except at the cache encoder. The decoders’ objective is to recover a function of the source and the side information. By viewing cache aided networks in terms of a general distributed source coding problem and through information theoretic arguments, we present inner and outer bounds on the fundamental tradeoff of cache memory size and update rate. Then, we specialize our general inner and outer bounds to a specific model of content delivery networks: file selection networks, in which the database is a collection of independent equal-size files and each user requests one of the files independently. For file selection networks, we provide an outer bound and two inner bounds (for centralized and decentralized caching strategies). For the case when the user request information is uniformly distributed, we characterize the rate versus cache size tradeoff to within a multiplicative gap of 4. By further extending our arguments to the framework of Maddah-Ali and Niesen, we also establish a new outer bound and two new inner bounds in which it is shown to recover the centralized and decentralized strategies, previously established by Maddah-Ali and Niesen. Finally, in terms of rate versus cache size tradeoff, we improve the previous multiplicative gap of 72 to 4.7 for the average case with uniform requests. [ABSTRACT FROM PUBLISHER]

A factor-graph representation of quantum-mechanical probabilities (involving any number of measurements) is proposed. Unlike standard statistical models, the proposed representation uses auxiliary variables (state variables) that are not random variables. All joint probability distributions are marginals of some complex-valued function $q$ , and it is demonstrated how the basic concepts of quantum mechanics relate to factorizations and marginals of $q$ . [ABSTRACT FROM AUTHOR]

We consider the problem of detecting abrupt parameter changes in a stochastic regression with unknown noise distribution. The process changes at some unknown point of time. Under general conditions on the regression function and unknown distributions of observations before and after the disruption, the paper develops a nonparametric cumulative sum procedure (CUSUM). Unlike likelihood-based CUSUM algorithms, constructed mostly on log-likelihood ratio statistics, we use a special system of basic statistics in Page’s procedure. By applying a sequential sampling scheme, which measures time in terms of accumulated Kullback–Leibler (K-L) divergence, we come to a system of statistics with the martingale properties close to those of the log-likelihood ratios. The proposed approach suggests also an alternative performance criterion in the analysis of the procedure by replacing the expected detection delay by the corresponding K-L divergence. We show that, under the false alarm probability constraint, the nonparametric CUSUM rule is optimal in the sense that it ensures the logarithmic asymptotic for the detection delay. [ABSTRACT FROM AUTHOR]

The objective of a genome-wide association study (GWAS) is to associate subsequences of individuals’ genomes to the observable characteristics called phenotypes (e.g., high blood pressure). Motivated by the GWAS problem, in this paper we introduce the information-theoretic problem of associated subsequence retrieval, where a dataset of N (possibly high-dimensional) sequences of length G, and their corresponding observable (binary) characteristics is given. The sequences are chosen independently and uniformly at random from $\mathcal {X}^{\text {G}}$ , where $\mathcal {X}$ is a finite alphabet. The observable (binary) characteristic is only related to a specific unknown subsequence of length $L$ of the sequences, called associated subsequence. For each sequence, if the associated subsequence of it belongs to a universal finite set, then it is more likely to display the observable characteristic (i.e., it is more likely that the observable characteristic is one). The goal is to retrieve the associated subsequence using a dataset of N sequences and their observable characteristics. We demonstrate that as the parameters N, G, and L grow, a threshold effect appears in the curve of probability of error versus the rate which is defined as ${{\it\text { Gh}}(\text {L}/\text {G})}/{\text {N}}$ , where $\text {h}(\cdot)$ is the binary entropy function. This effect allows us to define the capacity of associated subsequence retrieval. We develop an achievable scheme and a matching converse for this problem, and thus characterize its capacity in two scenarios: the zero-error-rate and the $\epsilon $ -error-rate. [ABSTRACT FROM AUTHOR]

The support recovery problem consists of determining a sparse subset of variables that is relevant in generating a set of observations. In this paper, we study the support recovery problem in the phase retrieval model consisting of noisy phaseless measurements, which arises in a diverse range of settings such as optical detection, X-ray crystallography, electron microscopy, and coherent diffractive imaging. Our focus is on information-theoretic fundamental limits under an approximate recovery criterion, considering both discrete and Gaussian models for the sparse non-zero entries, along with Gaussian measurement matrices. In both cases, our bounds provide sharp thresholds with near-matching constant factors in several scaling regimes on the sparsity and signal-to-noise ratio. As a key step towards obtaining these results, we develop new concentration bounds for the conditional information content of log-concave random variables, which may be of independent interest. [ABSTRACT FROM AUTHOR]

The classical problem of supervised learning is to infer an accurate estimate of a target variable $Y$ from a measured variable $X$ using a set of labeled training samples. Motivated by the increasingly distributed nature of data and decision making, this paper considers a variation of this classical problem in which the inference is distributed between two nodes, e.g., a mobile device and a cloud, with a rate constraint on the communication between them. The mobile device observes $X$ and sends a description $M$ of $X$ to the cloud, which computes an estimate $\hat {Y}$ of $Y$. We follow the recent minimax learning approach to study this inference problem and show that it corresponds to a one-shot minimax noisy lossy source coding problem. We then establish information theoretic bounds on the risk-rate Lagrangian cost, leading to a general method for designing a near-optimal descriptor-estimator pair. A key ingredient in the proof of our result is a refined version of the strong functional representation lemma previously used to establish several one-shot source coding theorems. Our results show that a naive estimate-compress scheme for rate-constrained inference is not optimal in general. When the distribution of $(X,Y)$ is known and the error is measured by the logarithmic loss, our bounds on the risk-rate Lagrangian cost provide a new one-shot operational interpretation of the information bottleneck. We also demonstrate a way to bound the excess risk of the descriptor-estimator pair obtained by our method. [ABSTRACT FROM AUTHOR]

We consider the recursive estimation of the probability density function of continuous random variables from a strongly mixing random sample. We revisit here earlier research on this subject by considering a more general class of recursive estimators, including the usual ones. We derive the quadratic mean error of the considered class of estimators. Moreover, we establish a central limit theorem by using Lindeberg’s method resulting in a simplification of the existing assumptions on the sequence of smooth parameters and the mixing coefficient. This is the main contribution of this paper. Finally, the feasibility of the proposed estimator is illustrated throughout an empirical study. [ABSTRACT FROM AUTHOR]

Algebraic network information theory is an emerging facet of network information theory, studying the achievable rates of random code ensembles that have algebraic structure, such as random linear codes. A distinguishing feature is that linear combinations of codewords can sometimes be decoded more efficiently than codewords themselves. The present work further develops this framework by studying the simultaneous decoding of multiple messages. Specifically, consider a receiver in a multi-user network that wishes to decode several messages. Simultaneous joint typicality decoding is one of the most powerful techniques for determining the fundamental limits at which reliable decoding is possible. This technique has historically been used in conjunction with random i.i.d. codebooks to establish achievable rate regions for networks. Recently, it has been shown that, in certain scenarios, nested linear codebooks in conjunction with “single-user” or sequential decoding can yield better achievable rates. For instance, the compute–forward problem examines the scenario of recovering $L \le K$ linear combinations of transmitted codewords over a $K$ -user multiple-access channel (MAC), and it is well established that linear codebooks can yield higher rates. This paper develops bounds for simultaneous joint typicality decoding used in conjunction with nested linear codebooks, and applies them to obtain a larger achievable region for compute–forward over a $K$ -user discrete memoryless MAC. The key technical challenge is that competing codeword tuples that are linearly dependent on the true codeword tuple introduce statistical dependencies, which requires careful partitioning of the associated error events. [ABSTRACT FROM AUTHOR]

In the distributed function computation problem, dichotomy theorems, initiated by Han-Kobayashi, seek to classify functions by whether the rate regions for function computation improve on the Slepian-Wolf regions or not. In this paper, we develop a general approach to derive converse bounds on the distributed function computation problem. By using this approach, we recover the sufficiency part, i.e. the conditions such that the Slepian-Wolf regions become optimal, of the known dichotomy theorems in the two-terminal distributed computing. Furthermore, we derive an improved sufficient condition on the dichotomy theorem in the multiterminal distributed computing for the class of i.i.d. sources with the positivity condition. Finally, we derive the matching sufficient and necessary condition on the dichotomy theorem in the multiterminal distributed computing for the class of smooth sources. [ABSTRACT FROM AUTHOR]

Proving or disproving an information inequality is a crucial step in establishing the converse results in coding theorems. However, an information inequality involving more than a few random variables is difficult to be proved or disproved manually. In 1997, Yeung developed a framework that uses linear programming for verifying linear information inequalities. Under the framework, this paper considers a few other problems that can be solved by using Lagrange duality and convex approximation. We will demonstrate how linear programming can be used to find an analytic proof of an information inequality or an analytic counterexample to disprove it if the inequality is not true in general. The way to automatically find a shortest proof or a smallest counterexample is explored. When a given information inequality cannot be proved, the sufficient conditions for a counterexample to disprove the information inequality are found by linear programming. Lastly, we propose a scalable algorithmic framework based on the alternating direction method of multipliers to accelerate solving a multitude of user-specific problems whose overall computational cost can be amortized with the number of users, and present its publicly-available software implementation for large-scale problems. [ABSTRACT FROM AUTHOR]

The capacity-achieving input distribution of the discrete-time, additive white Gaussian noise (AWGN) channel with an amplitude constraint is discrete and seems difficult to characterize explicitly. A dual capacity expression is used to derive analytic capacity upper bounds for scalar and vector AWGN channels. The scalar bound improves on McKellips’ bound and is within 0.1 bit of capacity for all signal-to-noise ratios (SNRs). The 2-D bound is within 0.15 bits of capacity provably up to 4.5 dB; numerical evidence suggests a similar gap for all SNRs. As the SNR tends to infinity, these bounds are accurate and match with a volume-based lower bound. For the 2-D complex case, an analytic lower bound is derived by using a concentric constellation and is shown to be within 1 bit of capacity. [ABSTRACT FROM PUBLISHER]

Testing the independence of the entries of multidimensional Gaussian observations is a very important problem in statistics, with a number of applications in signal processing, radar, cognitive radio, seismography, and multiple other fields. Typically, the problem is formulated as a binary hypothesis test, whereby the presence of correlation is declared when the value of a certain statistic is higher than a certain predetermined threshold. Most of the statistics for correlation tests are constructed from the sample correlation matrix (also known as sample coherence matrix in signal processing), which is defined as a power-normalized version of the sample covariance matrix. In this paper, correlation tests constructed from linear spectral statistics (LSS) of the sample correlation matrix are analyzed under the asymptotic framework where both sample size and observation dimension become large but comparable in magnitude. A central limit theorem (CLT) is established on this class of statistics, which is valid for generally correlated Gaussian observations. Results show that LSS asymptotically fluctuate as Gaussian random variables under both the hypotheses, with an asymptotic mean and variance that can be established for each particular test. In particular, this general CLT can be used to establish the asymptotic behavior of two of the most important correlation test statistics, namely the generalized likelihood ratio test and the Frobenius norm test, under both null and alternative hypotheses. As a by-product, it is established that LSS of sample covariance and sample correlation matrices have exactly the same first order behavior, but quite different asymptotic fluctuations in the second-order regime. In both the cases, the LSS asymptotically behave as Gaussian random variables, although with quite different asymptotic means and variances. [ABSTRACT FROM PUBLISHER]

This paper introduces a convenient strategy for coding and predicting sequences of independent, identically distributed random variables generated from a large alphabet of size $m$ . In particular, the size of the sample is allowed to be variable. The employment of a Poisson model and tilting method simplifies the implementation and analysis through independence. The resulting strategy is optimal within the class of distributions satisfying a moment condition, and it is close to optimal for the class of all i.i.d distributions on strings of a given length. The method also can be used to code and predict strings with a condition on the tail of the ordered counts, and it can be applied to distributions in an envelope class. Moreover, we show that our model permits exact computation of the minimax optimal code, for all alphabet sizes, when conditioning on the size of the sample. [ABSTRACT FROM PUBLISHER]

We consider random number conversion (RNC) through random number storage with restricted size. We clarify the relation between the performance of RNC and the size of storage in the framework of the first- and second-order asymptotics, and derive their rate regions. Then, we show that the results for RNC with restricted storage recover those for conventional RNC without storage in the limit of storage size. To treat RNC via restricted storage, we introduce a new kind of probability distributions named generalized Rayleigh-normal distributions. Using the generalized Rayleigh-normal distributions, we can describe the second-order asymptotic behavior of RNC via restricted storage in a unified manner. As an application to quantum information theory, we analyze LOCC conversion via entanglement storage with restricted size. Moreover, we derive the optimal LOCC compression rate under a constraint of conversion accuracy. [ABSTRACT FROM PUBLISHER]

In this paper, we study the problem of obtaining 1-of-2 string oblivious transfer (OT) between users Alice and Bob, in the presence of a passive eavesdropper Eve. The resource enabling OT in our setup is a noisy broadcast channel from Alice to Bob and Eve. Apart from the OT requirements between the users, Eve is not allowed to learn anything about the users’ inputs. When Alice and Bob are honest-but-curious and the noisy broadcast channel is made up of two independent binary erasure channels (connecting Alice–Bob and Alice–Eve), we derive the 1-of-2 string OT capacity for both 2-privacy (when Eve can collude with either Alice or Bob) and 1-privacy (when no such collusion is allowed). We generalize these capacity results to 1-of- $N$ string OT and study other variants of this problem. When Alice and/or Bob are malicious, we present a different scheme based on interactive hashing. This scheme is shown to be optimal for certain parameter regimes. We present a new formulation of multiple, simultaneous OTs between Alice–Bob and Alice–Cathy. For this new setup, we present schemes and outer bounds that match in all but one regime of parameters. Finally, we consider the setup where the broadcast channel is made up of a cascade of two independent binary erasure channels (connecting Alice–Bob and Bob–Eve) and 1-of-2 string OT is desired between Alice and Bob with 1-privacy. For this setup, we derive an upper and lower bound on the 1-of-2 string OT capacity which match in one of two possible parameter regimes. [ABSTRACT FROM PUBLISHER]

In this paper, we revisit the discrete lossy Gray-Wyner problem. In particular, we derive its optimal second-order coding rate region, its error exponent (reliability function), and its moderate deviations constant under mild conditions on the source. To obtain the second-order asymptotics, we extend some ideas from Watanabe’s work. In particular, we leverage the properties of an appropriate generalization of the conditional distortion-tilted information density, which was first introduced by Kostina and Verdú. The converse part uses a perturbation argument by Gu and Effros in their strong converse proof of the discrete Gray–Wyner problem. The achievability part uses two novel elements: 1) a generalization of various type covering lemmas and 2) the uniform continuity of the conditional rate-distortion function in both the source (joint) distribution and the distortion level. To obtain the error exponent, for the achievability part, we use the same generalized type covering lemma, and for the converse, we use the strong converse together with a change-of-measure technique. Finally, to obtain the moderate deviations constant, we apply the moderate deviations theorem to probabilities defined in terms of information spectrum quantities. [ABSTRACT FROM PUBLISHER]

An information theoretical analysis of nonadaptive group testing schemes based on sparse pooling graphs is presented. A pooling graph is a bipartite graph for which the adjacency matrix is a pooling matrix. The binary status of the objects to be tested are modeled by independent identically distributed. Bernoulli random variables with probability $p$ . An $(l, r, n)$ -regular pooling graph is a bipartite graph in which the left nodes have degree $l$ , the right nodes have degree $r$ , and $n$ is the number of left nodes. The main contribution of this paper is a direct coding theorem that gives the conditions for the existence of an estimator that can achieve an arbitrarily small probability of error. An estimator is a function that uses observations to infer the state of an object. The direct coding theorem is proved by averaging the upper bound on the probability of the estimation error of the typical set estimator over an $(l, r, n)$ -regular pooling graph ensemble. Numerical results indicate sharp threshold behaviors in the asymptotic regime. These results can provide a concrete benchmark for nonadaptive group testing of existing and emerging detection algorithms over a noiseless system. [ABSTRACT FROM PUBLISHER]

Treating interference as noise (TIN) as the most appropriate approach in dealing with interference and the conditions on its optimality has attracted the interest of researchers recently. However, our knowledge on necessary and sufficient conditions of TIN is restricted to a few setups with limited number of users. In this paper, we study the optimality of TIN in terms of the generalized degrees of freedom (GDoF) for a fundamental network, namely, the $M\times 2$ X-channel. To this end, the achievable GDoF of TIN with power allocations at the transmitters is studied. It turns out that the transmit power allocation maximizing the achievable GDOF is given by on–off signaling as long as the receivers use TIN. This leads to two variants of TIN, namely, P2P-TIN and 2-IC-TIN. While in the first variant the $M\times 2$ X-channel is reduced to a point-to-point (P2P) channel, in the second variant, the setup is reduced to a two-user interference channel in which the receivers use TIN. The optimality of these two variants is studied separately. To this end, novel genie-aided upper bounds on the capacity of the X-channel are established. The conditions on the optimality of P2P-TIN can be summarized as follows. P2P-TIN is GDoF-optimal if there exists a dominant multiple access channel or a dominant broadcast channel embedded in the X channel. Furthermore, the necessary and sufficient conditions on the GDoF-optimality of 2-IC-TIN are presented. Interestingly, it turns out that operating the $M\times 2$ X-channel in the 2-IC-TIN mode might be still GDOF optimal, although the conditions given by Geng et al. are violated. However, 2-IC-TIN is sub-optimal if there exists a single interferer which causes sufficiently strong interference at both receivers. The comparison of the results with the state of the art shows that the GDOF optimality of TIN is expanded significantldy. [ABSTRACT FROM PUBLISHER]

The impact of cooperation on interference management is investigated by studying an elemental wireless network, the so-called symmetric interference relay channel (IRC), from a generalized degrees of freedom (GDoF) perspective. This is motivated by the fact that the deployment of relays is considered as a remedy to overcome the bottleneck of current systems in terms of achievable rates. The focus of this paper is on the regime in which the interference link is weaker than the source-relay link in the IRC. Our approach toward studying the GDoF goes through the capacity analysis of the linear deterministic IRC (LD-IRC). New upper bounds on the sum capacity of the LD-IRC based on genie-aided approaches are established. These upper bounds together with some existing upper bounds are achieved by using four novel transmission schemes. Extending the upper bounds and the transmission schemes to the Gaussian case, the GDoF of the Gaussian IRC is characterized for the aforementioned regime. This completes the GDoF results available in the literature for the symmetric GDoF. It turns out that even if the incoming and outgoing links of the relay are both weaker than the desired channel, involving a relay can increase the GDoF. Interestingly, utilizing the relay in this case can increase the slope of the GDoF from −2 [in the interference channel (IC)] to −1 or 0. This shrinks the regime where ignoring the interference by treating it as noise is optimal. Furthermore, the analysis shows that if the relay ingoing and outgoing links are sufficiently strong, the relay is able to neutralize the interference completely. In this case, the bottleneck of the transmission will be the interference links, and hence, the GDoF increases if the interference link gets stronger. It is shown that in the strong interference regime, in contrast to the IC, the GDoF can be a monotonically decreasing function of the interference level. [ABSTRACT FROM PUBLISHER]

This paper gives improved Rényi entropy power inequalities (R-EPIs). Consider a sum Sn = \sum k=1^{n} Xk of n independent continuous random vectors taking values on {\mathbb {R}}^{d} , and let \alpha \in [1, \infty ] . An R-EPI provides a lower bound on the order- \alpha Rényi entropy power of S_{n} that, up to a multiplicative constant (which may depend in general on n, \alpha , d$ ), is equal to the sum of the order- \alpha $ Rényi entropy powers of the n$ random vectors . For $\alpha =1$ , the R-EPI coincides with the well-known entropy power inequality by Shannon. The first improved R-EPI is obtained by tightening the recent R-EPI by Bobkov and Chistyakov, which relies on the sharpened Young’s inequality. A further improvement of the R-EPI also relies on convex optimization and results on rank-one modification of a real-valued diagonal matrix. [ABSTRACT FROM PUBLISHER]

Let N be a random variable distributed according to some appropriate distribution over the set of products of two primes, such that factoring N is believed to be hard. The RSA assumption states that, given an a chosen uniformly at random from and an e \in {\mathbb {N}} \setminus \{1\} such that \gcd (e, \phi (N)) = 1 , it is computationally hard to find an x\in \mathbb Z N such that x^{e} - a \equiv 0 \pmod N . When complexity-theoretic (relative) lower bounds for certain cryptographic problems in a general model of computation seem to elude discovery, a common practice in cryptography is to give proofs of computational security in meaningful restricted models of computation. An example of such a restricted model that is interesting in cryptography is the generic group model that has been used for proving lower bounds for the discrete logarithm problem and other related problems. A generic model captures that an algorithm does not exploit the bit representation of the elements other than for testing equality. In this paper, we prove that the problem of factoring $N$ can be efficiently reduced to solving the RSA problem on {\mathbb {Z}}_{N} in the generic ring model of computation, where an algorithm can perform ring operations, inverse ring operations, and test equality. This provides evidence toward the soundness of the RSA encryption and digital signature scheme, in particular showing that under the factoring assumption, they are not vulnerable to certain kinds of cryptanalytic attacks. [ABSTRACT FROM AUTHOR]

Let \mathcal N=\1,\ldots ,n\ . The entropy function h of a set of n discrete random variables \{X_{i}:i\in \mathcal {N} \} is a 2^{n} -dimensional vector whose entries are \text h( \mathcal A)\triangleq H(X \mathcal A) , \mathcal A\subset \mathcal {N} , the (joint) entropies of the subsets of the set of n random variables with H(X_\emptyset )=0 by convention. The set of all entropy functions for n discrete random variables, denoted by , is called the entropy region for n . Characterization of \Gamma ^*n and its closure \overline {\Gamma ^*n} are well-known open problems in information theory. They are important not only because they play key roles in information theory problems but also they are related to other subjects in mathematics and physics. In this paper, we consider partition-symmetrical entropy functions. Let p=\ \mathcal N1,\ldots , \mathcal {Nt\} be a t -partition of \mathcal {N} . An entropy function \bf h is called p -symmetrical if for all {\mathcal{ A}},{\mathcal{ B}} \subset {\mathcal{ N}} , \text {h}({\mathcal{ A}}) = \text {h}({\mathcal{ B}}) whenever \mathcal A \cap \mathcal Ni = {\mathcal{ B}} \cap \mathcal Ni , i = 1, \ldots ,t . The set of all the p -symmetrical entropy functions, denoted by \Psi ^{*}_{p} , is called p -symmetrical entropy function region. We prove that \overline \Psi ^*p , the closure of \Psi ^{*}p , is completely characterized by Shannon-type information inequalities if and only if $p$ is the 1-partition or a 2-partition with one of its blocks being a singleton. The characterization of the partition-symmetrical entropy functions can be useful for solving some information theory and related problems where symmetry exists in the structure of the problems. [ABSTRACT FROM AUTHOR]

Unveiling a fundamental link between information theory and estimation theory, the I-MMSE relationship by Guo et al., together with its numerous extensions, has great theoretical significance and various practical applications. On the other hand, its influences to date have been restricted to channels without feedback or memory, due to the absence of its extensions to such channels. In this paper, we propose the extensions of the I-MMSE relationship to discrete-time and continuous-time Gaussian channels with feedback and/or memory. Our approach is based on a very simple observation, which can be applied to other scenarios, such as a simple and direct proof of the classical de Bruijn’s identity. [ABSTRACT FROM AUTHOR]

The best known inner bound for the two-user discrete memoryless interference channel is the Han–Kobayashi rate region. The coding schemes that achieve this region are based on rate-splitting and superposition coding. In this paper, we develop a multicoding scheme to achieve the same rate region. A key advantage of the multicoding nature of the proposed coding scheme is that it can be naturally extended to more general settings, such as when encoders have state information or can overhear each other. In particular, we extend our coding scheme to characterize the capacity region of the state-dependent deterministic Z-interference channel when noncausal state information is available at the interfering transmitter. We specialize our results to the case of the linear deterministic model with ON/OFF interference, which models a wireless system where a cognitive transmitter is noncausally aware of the times it interferes with a primary transmission. For this special case, we provide an explicit expression for the capacity region and discuss some interesting properties of the optimal strategy. We also extend our multicoding scheme to find the capacity region of the deterministic Z-interference channel when the signal of the interfering transmitter can be overheard at the other transmitter (also known as unidirectional partial cribbing). [ABSTRACT FROM PUBLISHER]

This paper extends the empirical minimum divergence approach for models, which satisfy linear constraints with respect to the probability measure of the underlying variable (moment constraints) to the case where such constraints pertain to its quantile measure (called here semiparametric quantile models). The case when these constraints describe shape conditions as handled by the L-moments is considered, and both the description of these models as well as the resulting nonclassical minimum divergence procedures are presented. These models describe neighbourhoods of classical models used mainly for their tail behavior, for example, neighborhoods of Pareto or Weibull distributions, with which they may share the same first L-moments. The properties of the resulting estimators are illustrated by simulated examples comparing maximum likelihood estimators on Pareto and Weibull models to the minimum chi-square empirical divergence approach on semiparametric quantile models, and others. [ABSTRACT FROM PUBLISHER]

Phase retrieval refers to algorithmic methods for recovering a signal from its phaseless measurements. There has been recent interest in understanding the performance of local search algorithms that work directly on the non-convex formulation of the problem. Due to the non-convexity of the problem, the success of these local search algorithms depends heavily on their starting points. The most widely used initialization scheme is the spectral method, in which the leading eigenvector of a data-dependent matrix is used as a starting point. Recently, the performance of the spectral initialization was characterized accurately for measurement matrices with independent and identically distributed entries. This paper aims to obtain the same level of knowledge for isotropically random column-orthogonal matrices, which are substantially better models for practical phase retrieval systems. Towards this goal, we consider the asymptotic setting in which the number of measurements $m$ , and the dimension of the signal, $n$ , diverge to infinity with $m/n = \delta \in (1,\infty)$ , and obtain a simple expression for the overlap between the spectral estimator and the true signal vector. [ABSTRACT FROM AUTHOR]

This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating $P\ast \mathcal {N}_\sigma $ , for $\mathcal {N}_\sigma \triangleq \mathcal {N}(0,\sigma ^{2} \mathrm {I}_{d})$ , by $\hat {P}_{n}\ast \mathcal {N}_\sigma $ under different statistical distances, where $\hat {P}_{n}$ is the empirical measure. We examine the convergence in terms of the Wasserstein distance, total variation (TV), Kullback-Leibler (KL) divergence, and $\chi ^{2}$ -divergence. We show that the approximation error under the TV distance and 1-Wasserstein distance ($\mathsf {W}_{1}$) converges at the rate $e^{O(d)} n^{-1/2}$ in remarkable contrast to a (typical) $n^{-\frac {1}{d}}$ rate for unsmoothed $\mathsf {W}_{1}$ (and $d\ge 3$). Similarly, for the KL divergence, squared 2-Wasserstein distance ($\mathsf {W}_{2}^{2}$), and $\chi ^{2}$ -divergence, the convergence rate is $e^{O(d)} n^{-1}$ , but only if $P$ achieves finite input-output $\chi ^{2}$ mutual information across the additive white Gaussian noise (AWGN) channel. If the latter condition is not met, the rate changes to $\omega \left ({n^{-1}}\right)$ for the KL divergence and $\mathsf {W}_{2}^{2}$ , while the $\chi ^{2}$ -divergence becomes infinite – a curious dichotomy. As an application we consider estimating the differential entropy $h(S+Z)$ , where $S\sim P$ and $Z\sim \mathcal {N}_\sigma $ are independent $d$ -dimensional random variables. The distribution $P$ is unknown and belongs to some nonparametric class, but n independently and identically distributed (i.i.d) samples from it are available. Despite the regularizing effect of noise, we first show that any good estimator (within an additive gap) for this problem must have a sample complexity that is exponential in d. We then leverage the above empirical approximation results to show that the absolute-error risk of the plug-in estimator converges as $e^{O(d)} n^{-1/2}$ , thus attaining the parametric rate in n. This establishes the plug-in estimator as minimax rate-optimal for the considered problem, with sharp dependence of the convergence rate both in n and d. We provide numerical results comparing the performance of the plug-in estimator to that of general-purpose (unstructured) differential entropy estimators (based on kernel density estimation (KDE) or k nearest neighbors (kNN) techniques) applied to samples of $S+Z$. These results reveal a significant empirical superiority of the plug-in to state-of-the-art KDE and kNN methods. As a motivating utilization of the plug-in approach, we estimate information flows in deep neural networks and discuss Tishby’s Information Bottleneck and the compression conjecture, among others. [ABSTRACT FROM AUTHOR]

Recently, two extensions of Wyner’s common information—exact and Rényi common informations—were introduced respectively by Kumar, Li, and El Gamal (KLE), and the present authors. The class of common information problems involves determining the minimum rate of the common input to two independent processors needed to exactly or approximately generate a target joint distribution. For the exact common information problem, exact generation of the target distribution is required, while for Wyner’s and $\alpha $ -Rényi common informations, the relative entropy and Rényi divergence with order $\alpha $ were respectively used to quantify the discrepancy between the synthesized and target distributions. The exact common information is larger than or equal to Wyner’s common information. However, it was hitherto unknown whether the former is strictly larger than the latter for some joint distributions. In this paper, we first establish the equivalence between the exact and $\infty $ -Rényi common informations, and then provide single-letter upper and lower bounds for these two quantities. For doubly symmetric binary sources, we show that the upper and lower bounds coincide, which implies that for such sources, the exact and $\infty $ -Rényi common informations are completely characterized. Interestingly, we observe that for such sources, these two common informations are strictly larger than Wyner’s. This answers an open problem posed by KLE. Furthermore, we extend Wyner’s, $\infty $ -Rényi, and exact common informations to sources with countably infinite or continuous alphabets, including Gaussian sources. [ABSTRACT FROM AUTHOR]