results for au:Tan_V in:cs

- We consider the variable-to-fixed length lossy source coding (VFSC) problem. The optimal compression rate of the average length of variable-to-fixed source coding, allowing a non-vanishing probability of excess-distortion $\varepsilon$, is shown to be equal to $(1-\varepsilon)R(D)$, where $R(D)$ is the rate-distortion function of the source. In comparison to the related results of Koga and Yamamoto as well as Kostina, Polyanskiy, and Verdú for fixed-to-variable length source coding, our results demonstrate an interesting feature that variable-to-fixed length source coding has the same first-order compression rate as fixed-to-variable length source coding.
- We derive exact (ensemble-tight) error and erasure exponents for the asymmetric broadcast channel given a random superposition codebook. We consider Forney's optimal decoder for both messages and the message pair for the receiver that decodes both messages. We prove that the optimal decoder designed to decode the pair of messages achieves the optimal trade-off between the total and undetected exponents associated with the optimal decoder for the private message. We propose convex optimization-based procedures to evaluate the exponents efficiently. Numerical examples are presented to illustrate the results.
- The zero-error capacity of state-dependent channels with noiseless feedback is determined, under the assumption that the transmitter and the receiver are allowed to use variable-length coding schemes. Various cases are analyzed, with the employed coding schemes having either bounded or unbounded codeword lengths and with state information revealed to the encoder and/or decoder in a strictly causal, causal, or non-causal manner. In each of these settings, necessary and sufficient conditions for positivity of the zero-error capacity are obtained and it is shown that, whenever the zero-error capacity is positive, it equals the conventional vanishing-error capacity. Moreover, it is shown that the vanishing-error capacity of state-dependent channels is not increased by the use of feedback and variable-length coding. A comparison of the results with the recently solved fixed-length case is also given.
- A coupling of two distributions $P_{X}$ and $P_{Y}$ is a joint distribution $P_{XY}$ with marginal distributions equal to $P_{X}$ and $P_{Y}$. Given marginals $P_{X}$ and $P_{Y}$ and a real-valued function $f(P_{XY})$ of the joint distribution $P_{XY}$, what is its minimum over all couplings $P_{XY}$ of $P_{X}$ and $P_{Y}$? We study the asymptotics of such coupling problems with different $f$'s. These include the maximal coupling, minimum distance coupling, maximal guessing coupling, and minimum entropy coupling problems. We characterize the limiting values of these coupling problems as the number of copies of $X$ and $Y$ tends to infinity. We show that they typically converge at least exponentially fast to their limits. Moreover, for the problems of maximal coupling and minimum excess-distance probability coupling, we also characterize (or bound) the optimal convergence rates (exponents). Furthermore, for the maximal guessing coupling problem we show that it is equivalent to the probability distribution approximation problem. Therefore, some existing results the latter problem can be used to derive the asymptotics of the maximal guessing coupling problem. We also study the asymptotics of the maximal guessing coupling problem for two \emphgeneral sources and a generalization of this problem, named the \emphmaximal guessing coupling through a channel problem. We apply the preceding results to several new information-theoretic problems, including exact intrinsic randomness, exact resolvability, channel capacity with input distribution constraint, and perfect stealth and secrecy communication.
- We consider a universal joint source channel coding (JSCC) scheme to transmit an arbitrary memoryless source over an arbitrary additive channel. We adopt an architecture that consists of Gaussian codebooks for both the source reproduction sequences and channel codewords. The natural minimum Euclidean distance encoder and decoder, however, need to be judiciously modified to ensure universality as well as to obtain the best (highest) possible communication rates. In particular, we consider the analogue of an unequal error (or message) protection scheme in which all sources are partitioned into disjoint power type classes. We also regularize the nearest neighbor decoder so an appropriate measure of the size of each power type class is taken into account in the decoding strategy. For such an architecture, we derive ensemble tight second-order and moderate deviations results. Our first-order result generalizes seminal results by Lapidoth (1996, 1997). The dispersion of our JSCC scheme is a linear combination of the mismatched dispersions for the channel coding saddle-point problem by Scarlett, Tan and Durisi (2017) and the rate-distortion saddle-point problem by the present authors, thus also generalizing these results.
- We consider a covert communication scenario where a legitimate transmitter wishes to communicate simultaneously to two legitimate receivers while ensuring that the communication is not detected by an adversary, also called the warden. The legitimate receivers and the adversary observe the transmission from the legitimate transmitter via a three-user discrete or Gaussian memoryless broadcast channel. We focus on the case where the "no-input" symbol is not redundant, i.e., the output distribution at the warden induced by the no-input symbol is not a mixture of the output distributions induced by other input symbols, so that the covert communication is governed by the square root law, i.e., at most $\Theta(\sqrt{n})$ bits can be transmitted over $n$ channel uses. We show that for such a setting of covert communication over broadcast channels, a simple time-division strategy achieves the optimal throughputs. Our result implies that a code that uses two separate optimal point-to-point codes each designed for the constituent channels and each used for a fraction of the time is optimal in the sense that it achieves the best constants of the $\sqrt{n}$-scaling for the throughputs. Our proof strategy combines several elements in the network information theory literature, including concave envelope representations of the capacity regions of broadcast channels and El Gamal's outer bound for more capable broadcast channels.
- We consider transmission of discrete memoryless sources (DMSes) across discrete memoryless channels (DMCs) using variable-length lossy source-channel codes with feedback. The reliability function (optimum error exponent) is shown to be equal to $\max\{0, B(1-R(D)/C)\},$ where $R(D)$ is the rate-distortion function of the source, $B$ is the maximum relative entropy between output distributions of the DMC, and $C$ is the Shannon capacity of the channel. We show that, in this setting and in this asymptotic regime, separate source-channel coding is, in fact, optimal.
- We consider the problem of aggregating pairwise comparisons to obtain a consensus ranking order over a collection of objects. We use the popular Bradley-Terry-Luce (BTL) model which allows us to probabilistically describe pairwise comparisons between objects. In particular, we employ the Bayesian BTL model which allows for meaningful prior assumptions and to cope with situations where the number of objects is large and the number of comparisons between some objects is small or even zero. For the conventional Bayesian BTL model, we derive information-theoretic lower bounds on the Bayes risk of estimators for norm-based distortion functions. We compare the information-theoretic lower bound with the Bayesian Cramér-Rao lower bound we derive for the case when the Bayes risk is the mean squared error. We illustrate the utility of the bounds through simulations by comparing them with the error performance of an expectation-maximization based inference algorithm proposed for the Bayesian BTL model. We draw parallels between pairwise comparisons in the BTL model and inter-player games represented as edges in a comparison graph and analyze the effect of various graph structures on the lower bounds. We also extend the information-theoretic and Bayesian Cramér-Rao lower bounds to the more general Bayesian BTL model which takes into account home-field advantage.
- We study a generalized version of Wyner's common information problem (also coined the distributed sources simulation problem). The original common information problem consists in understanding the minimum rate of the common input to independent processors to generate an approximation of a joint distribution when the distance measure used to quantify the discrepancy between the synthesized and target distributions is the normalized relative entropy. Our generalization involves changing the distance measure to the unnormalized and normalized Rényi divergences of order $\alpha=1+s\in[0,2]$. We show that the minimum rate needed to ensure the Rényi divergences between the distribution induced by a code and the target distribution vanishes remains the same as the one in Wyner's setting, except when the order $\alpha=1+s=0$. This implies that Wyner's common information is rather robust to the choice of distance measure employed. As a by product of the proofs used to the establish the above results, the exponential strong converse for the common information problem under the total variation distance measure is established.
- The rate-distortion saddle-point problem considered by Lapidoth (1997) consists in finding the minimum rate to compress an arbitrary ergodic source when one is constrained to use a random Gaussian codebook and minimum (Euclidean) distance encoding is employed. We extend Lapidoth's analysis in several directions in this paper. Firstly, we consider refined asymptotics. In particular, when the source is stationary and memoryless, we establish the second-order, moderate, and large deviation asymptotics of the problem. Secondly, by "random Gaussian codebook", Lapidoth refers to a collection of random codewords, each of which is drawn independently and uniformly from the surface of an $n$-dimensional sphere. To be more precise, we term this as a spherical Gaussian codebook. We also consider i.i.d.\ Gaussian codebooks in which each random codeword is drawn independently from a product Gaussian distribution. We derive the second-order, moderate, and large deviation asymptotics when i.i.d.\ Gaussian codebooks are employed. Interestingly, in contrast to the recent work on the channel coding counterpart by Scarlett, Tan and Durisi (2017), the dispersions for spherical and i.i.d.\ Gaussian codebooks are identical. The optimal excess-distortion exponents for both spherical and i.i.d. Gaussian codebooks are established for all rates. Furthermore, we prove that the i.i.d. Gaussian codebook has a strictly larger excess-distortion exponent than the spherical counterpart for any rate larger the first order coding rate.
- We consider data transmission across discrete memoryless channels (DMCs) using variable-length codes with feedback. We consider the family of such codes whose rates are $\rho_N$ below the channel capacity $C$, where $\rho_N$ is a positive sequence that tends to zero slower than the reciprocal of the square root of the expectation of the (random) blocklength $N$. This is known as the moderate deviations regime and we establish the optimal moderate deviations constant. We show that in this scenario, the error probability decays sub-exponentially with speed $\exp(-(B/C)N\rho_N)$, where $B$ is the maximum relative entropy between output distributions of the DMC.
- The conventional channel resolvability problem refers to the determination of the minimum rate needed for an input process to approximate the output distribution of a channel in either the total variation distance or the relative entropy. In contrast to previous works, in this paper, we use the (normalized or unnormalized) Rényi divergence (with the Rényi parameter in [0,2]) to measure the level of approximation. We also provide asymptotic expressions for normalized Rényi divergence when the Rényi parameter is larger than or equal to $1$ as well as (lower and upper) bounds for the case when the same parameter is smaller than $1$. We characterize the minimum rate needed to ensure that the Rényi resolvability vanishes asymptotically. The optimal rates are the same for both the normalized and unnormalized cases. In addition, the minimum rate when the Rényi parameter no larger than $1$ equals the minimum mutual information over all input distributions that induce the target output distribution similarly to the traditional case. When the Rényi parameter is larger than $1$ the minimum rate is, in general, larger than the mutual information. The optimal Rényi resolvability is proven to vanish at least exponentially fast for both of these two cases, as long as the code rate is larger than the minimum admissible one. The optimal exponential rate of decay for i.i.d. random codes is also characterized exactly. We apply these results to the wiretap channel, and completely characterize the optimal tradeoff between the rates of the secret and non-secret messages when the leakage measure is given by the (unnormalized) Rényi divergence. This tradeoff differs from the conventional setting when the leakage is measured by the traditional mutual information.
- An exact information spectrum-type formula for the maximum size of finite length block codes subject to a minimum pairwise distance constraint is presented. This formula can be applied to codes for a broad class of distance measures. As revealed by the formula, the largest code size is fully characterized by the information spectrum of the distance between two independent and identically distributed (i.i.d.) random codewords drawn from an optimal distribution. A new family of lower bounds to the maximal code size is thus established, and the well-known Gilbert-Varshamov (GV) lower bound is a special case of this family. By duality, an explicit expression for the largest minimum distance of finite length block codes of a fixed code size is also obtained. Under an arbitrary uniformly bounded symmetric distance measure, the asymptotic largest code rate (in the block length $n$) attainable for a sequence of $(n,M,n\delta)$-codes is given exactly by the maximum large deviation rate function of the normalized distance between two i.i.d. random codewords. The exact information spectrum-type formula also yields bounds on the second-order terms in the asymptotic expansion of the optimum finite length rate for block codes with a fixed normalized minimum distance.
- This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The scaling exponent $\mu$ of polar codes for a memoryless channel $q_{Y|X}$ with capacity $I(q_{Y|X})$ characterizes the closest gap between the capacity and non-asymptotic achievable rates in the following way: For a fixed $\varepsilon \in (0, 1)$, the gap between the capacity $I(q_{Y|X})$ and the maximum non-asymptotic rate $R_n^*$ achieved by a length-$n$ polar code with average error probability $\varepsilon$ scales as $n^{-1/\mu}$, i.e., $I(q_{Y|X})-R_n^* = \Theta(n^{-1/\mu})$. It is well known that the scaling exponent $\mu$ for any binary-input memoryless channel (BMC) with $I(q_{Y|X})\in(0,1)$ is bounded above by $4.714$, which was shown by an explicit construction of polar codes. Our main result shows that $4.714$ remains to be a valid upper bound on the scaling exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of $O(n^{-1/\mu}\sqrt{\log n})$ by using an input alphabet consisting of $n$ constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of $n$ constellations can be achieved within a gap of $O(n^{-1/\mu}\log n)$ by using a superposition of $\log n$ binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as $n$ grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel.
- Hypothesis testing is a statistical inference framework for determining the true distribution among a set of possible distributions for a given dataset. Privacy restrictions may require the curator of the data or the respondents themselves to share data with the test only after applying a randomizing privacy mechanism. This work considers mutual information (MI) as the privacy metric for measuring leakage. In addition, motivated by the Chernoff-Stein lemma, the relative entropy between pairs of distributions of the output (generated by the privacy mechanism) is chosen as the utility metric. For these metrics, the goal is to find the optimal privacy-utility trade-off (PUT) and the corresponding optimal privacy mechanism for both binary and m-ary hypothesis testing. Focusing on the high privacy regime, Euclidean information-theoretic approximations of the binary and m-ary PUT problems are developed. The solutions for the approximation problems clarify that an MI-based privacy metric preserves the privacy of the source symbols in inverse proportion to their likelihoods.
- We revisit the stochastic limited-memory BFGS (L-BFGS) algorithm. By proposing a new framework for the convergence analysis, we prove improved convergence rates and computational complexities of the stochastic L-BFGS algorithms compared to previous works. In addition, we propose several practical acceleration strategies to speed up the empirical performance of such algorithms. We also provide theoretical analyses for most of the strategies. Experiments on large-scale logistic and ridge regression problems demonstrate that our proposed strategies yield significant improvements vis-à-vis competing state-of-the-art algorithms.
- The learning of mixture models can be viewed as a clustering problem. Indeed, given data samples independently generated from a mixture of distributions, we often would like to find the correct target clustering of the samples according to which component distribution they were generated from. For a clustering problem, practitioners often choose to use the simple k-means algorithm. k-means attempts to find an optimal clustering which minimizes the sum-of-squared distance between each point and its cluster center. In this paper, we provide sufficient conditions for the closeness of any optimal clustering and the correct target clustering assuming that the data samples are generated from a mixture of log-concave distributions. Moreover, we show that under similar or even weaker conditions on the mixture model, any optimal clustering for the samples with reduced dimensionality is also close to the correct target clustering. These results provide intuition for the informativeness of k-means (with and without dimensionality reduction) as an algorithm for learning mixture models. We verify the correctness of our theorems using numerical experiments and demonstrate using datasets with reduced dimensionality significant speed ups for the time required to perform clustering.
- This paper investigates the asymptotic expansion for the maximum rate of fixed-length codes over a parallel Gaussian channel with feedback under the following setting: A peak power constraint is imposed on every transmitted codeword, and the average error probability of decoding the transmitted message is non-vanishing as the blocklength increases. It is well known that the presence of feedback does not increase the first-order asymptotics of the channel, i.e., capacity, in the asymptotic expansion, and the closed-form expression of the capacity can be obtained by the well-known water-filling algorithm. The main contribution of this paper is a self-contained proof of an upper bound on the second-order asymptotics of the parallel Gaussian channel with feedback. The proof techniques involve developing an information spectrum bound followed by using Curtiss' theorem to show that a sum of dependent random variables associated with the information spectrum bound converges in distribution to a sum of independent random variables, thus facilitating the use of the usual central limit theorem. Combined with existing achievability results, our result implies that the presence of feedback does not improve the second-order asymptotics.
- We revisit the high-dimensional content identification with lossy recovery problem (Tuncel and Gündüz, 2014) and establish an exponential strong converse theorem. As a corollary of the exponential strong converse theorem, we derive an upper bound on the joint identification-error and excess-distortion exponent for the problem. Our main results can be specialized to the biometrical identification problem~(Willems, 2003) and the content identification problem~(Tuncel, 2009) since these two problems are both special cases of the content identification with lossy recovery problem. We leverage the information spectrum method introduced by Oohama and adapt the strong converse techniques therein to be applicable to the problem at hand.
- The problem of publishing privacy-guaranteed data for hypothesis testing is studied using the maximal leakage (ML) as a metric for privacy and the type-II error exponent as the utility metric. The optimal mechanism (random mapping) that maximizes utility for a bounded leakage guarantee is determined for the entire leakage range for binary datasets. For non-binary datasets, approximations in the high privacy and high utility regimes are developed. The results show that, for any desired leakage level, maximizing utility forces the ML privacy mechanism to reveal partial to complete knowledge about a subset of the source alphabet. The results developed on maximizing a convex function over a polytope may also of an independent interest.
- We analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases. To characterise the fundamental tradeoff between decoding error, code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by Altug and Wagner as well as Polyanskiy and Verdu. We derive such a tradeoff for classical-quantum (as well as image-additive) channels in terms of the channel capacity and the channel dispersion, giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data. To derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime. Due to the central importance of the latter task, we expect that our techniques will find further applications in the analysis of other quantum information processing tasks.
- 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 the list size can be reduced to one and the original message can be recovered with high probability as the length of the code grows.
- This paper investigates the achievable rates of an additive white Gaussian noise (AWGN) energy-harvesting (EH) channel with an infinite battery. The EH process is characterized by a sequence of blocks of harvested energy, which is known causally at the source. The harvested energy remains constant within a block while the harvested energy across different blocks is characterized by a sequence of independent and identically distributed (i.i.d.) random variables. The blocks have length $L$, which can be interpreted as the coherence time of the energy arrival process. If $L$ is a constant or grows sublinearly in the blocklength $n$, we fully characterize the first-order term in the asymptotic expansion of the maximum transmission rate subject to a fixed tolerable error probability $\varepsilon$. The first-order term is known as the $\varepsilon$-capacity. In addition, we obtain lower and upper bounds on the second-order term in the asymptotic expansion, which reveal that the second order term scales as $\sqrt{\frac{L}{n}}$ for any $\varepsilon$ less than $1/2$. The lower bound is obtained through analyzing the save-and-transmit strategy. If $L$ grows linearly in $n$, we obtain lower and upper bounds on the $\varepsilon$-capacity, which coincide whenever the cumulative distribution function (cdf) of the EH random variable is continuous and strictly increasing. In order to achieve the lower bound, we have proposed a novel adaptive save-and-transmit strategy, which chooses different save-and-transmit codes across different blocks according to the energy variation across the blocks.
- This work investigates the limits of communication over a noisy channel that wears out, in the sense of signal-dependent catastrophic failure. In particular, we consider a channel that starts as a memoryless binary-input channel and when the number of transmitted ones causes a sufficient amount of damage, the channel ceases to convey signals. We restrict attention to constant composition codes. Since infinite blocklength codes will always wear out the channel for any finite threshold of failure and therefore convey no information, we make use of finite blocklength codes to determine the maximum expected transmission volume at a given level of average error probability. We show that this maximization problem has a recursive form and can be solved by dynamic programming. A discussion of damage state feedback in channels that wear out is also provided. Numerical results show that a sequence of block codes is preferred to a single block code for streaming sources.
- We derive upper and lower bounds on the reliability function for the common-message discrete memoryless broadcast channel with variable-length feedback. We show that the bounds are tight when the broadcast channel is stochastically degraded. For the achievability part, we adapt Yamamoto and Itoh's coding scheme by controlling the expectation of the maximum of a set of stopping times. For the converse part, we adapt Burnashev's proof techniques for establishing the reliability functions for (point-to-point) discrete memoryless channels with variable-length feedback and sequential hypothesis testing.
- Motivated by communication channels in which the transmitted sequences are subject to random permutations, as well as by certain DNA storage systems, we study the error control problem in settings where the information is stored/transmitted in the form of multisets of symbols from a given finite alphabet. A general channel model is assumed in which the transmitted multisets are potentially impaired by insertions, deletions, substitutions, and erasures of symbols. Several constructions of error-correcting codes for this channel are described, and bounds on the size of optimal codes correcting any given number of errors derived. The construction based on the notion of Sidon sets in finite Abelian groups is shown to be optimal, in the sense of the asymptotic scaling of code redundancy, for any "error radius" and any alphabet size. It is also shown to be optimal in the stronger sense of maximal code cardinality in various cases.
- We propose a geometric assumption on nonnegative data matrices such that under this assumption, we are able to provide upper bounds (both deterministic and probabilistic) on the relative error of nonnegative matrix factorization (NMF). The algorithm we propose first uses the geometric assumption to obtain an exact clustering of the columns of the data matrix; subsequently, it employs several rank-one NMFs to obtain the final decomposition. When applied to data matrices generated from our statistical model, we observe that our proposed algorithm produces factor matrices with comparable relative errors vis-à-vis classical NMF algorithms but with much faster speeds. On face image and hyperspectral imaging datasets, we demonstrate that our algorithm provides an excellent initialization for applying other NMF algorithms at a low computational cost. Finally, we show on face and text datasets that the combinations of our algorithm and several classical NMF algorithms outperform other algorithms in terms of clustering performance.
- Given a sufficient statistic for a parametric family of distributions, one can estimate the parameter without access to the data. However, the memory or code size for storing the sufficient statistic may nonetheless still be prohibitive. Indeed, for $n$ independent samples drawn from a $k$-nomial distribution with $d=k-1$ degrees of freedom, the length of the code scales as $d\log n+O(1)$. In many applications, we may not have a useful notion of sufficient statistics (e.g., when the parametric family is not an exponential family) and we also may not need to reconstruct the generating distribution exactly. By adopting a Shannon-theoretic approach in which we allow a small error in estimating the generating distribution, we construct various \em approximate sufficient statistics and show that the code length can be reduced to $\frac{d}{2}\log n+O(1)$. We consider errors measured according to the relative entropy and variational distance criteria. For the code constructions, we leverage Rissanen's minimum description length principle, which yields a non-vanishing error measured according to the relative entropy. For the converse parts, we use Clarke and Barron's formula for the relative entropy of a parametrized distribution and the corresponding mixture distribution. However, this method only yields a weak converse for the variational distance. We develop new techniques to achieve vanishing errors and we also prove strong converses. The latter means that even if the code is allowed to have a non-vanishing error, its length must still be at least $\frac{d}{2}\log n$.
- A $ B_h $ set (or Sidon set of order $ h $) in an Abelian group $ G $ is any subset $ \{b_0, b_1, \ldots,b_{n}\} $ of $ G $ with the property that all the sums $ b_{i_1} + \cdots + b_{i_h} $ are different up to the order of the summands. Let $ \phi(h,n) $ denote the order of the smallest Abelian group containing a $ B_h $ set of cardinality $ n + 1 $. It is shown that \[ \lim_h \to ∞ \frac \phi(h,n) h^n = \frac1n! \delta_L(\triangle^n) , \]where $ \delta_L(\triangle^n) $ is the lattice packing density of an $ n $-simplex in Euclidean space. This determines the asymptotics exactly in cases where this density is known ($ n \leq 3 $) and gives improved bounds on $ \phi(h,n) $ in the remaining cases. The corresponding geometric characterization of bases of order $ h $ in finite Abelian groups in terms of lattice coverings by simplices is also given.
- The multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical convergence guarantees have only been derived for a few special divergences without regularization. In this work, we provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and regularizers. Our main result shows the sequence of iterates (i.e., pairs of basis and coefficient matrices) produced by the MU algorithm converges to the set of stationary points of the non-convex NMF optimization problem. Our proof strategy has the potential to open up new avenues for analyzing similar problems in machine learning and signal processing.
- We characterize the fundamental limits of transmission of information over a Gaussian multiple access channel (MAC) with the use of variable-length feedback codes and under a non-vanishing error probability formalism. We develop new achievability and converse techniques to handle the continuous nature of the channel and the presence of expected power constraints. We establish the $\varepsilon$-capacity regions and bounds on the second-order asymptotics of the Gaussian MAC with variable-length feedback with termination (VLFT) codes and stop-feedback codes. We show that the former outperforms the latter significantly. Due to the multi-terminal nature of the channel model, we leverage tools from renewal theory developed by Lai and Siegmund to bound the asymptotic behavior of the maximum of a finite number of stopping times.
- Binary hypothesis testing under the Neyman-Pearson formalism is a statistical inference framework for distinguishing data generated by two different source distributions. Privacy restrictions may require the curator of the data or the data respondents themselves to share data with the test only after applying a randomizing privacy mechanism. Using mutual information as the privacy metric and the relative entropy between the two distributions of the output (postrandomization) source classes as the utility metric (motivated by the Chernoff-Stein Lemma), this work focuses on finding an optimal mechanism that maximizes the chosen utility function while ensuring that the mutual information based leakage for both source distributions is bounded. Focusing on the high privacy regime, an Euclidean information-theoretic (E-IT) approximation to the tradeoff problem is presented. It is shown that the solution to the E-IT approximation is independent of the alphabet size and clarifies that a mutual information based privacy metric preserves the privacy of the source symbols in inverse proportion to their likelihood.
- This paper considers a multimessage network where each node may send a message to any other node in the network. Under the discrete memoryless model, we prove the strong converse theorem for any network with tight cut-set bound, i.e., whose cut-set bound is achievable. Our result implies that for any network with tight cut-set bound and any fixed rate vector that resides outside the capacity region, the average error probabilities of any sequence of length-$n$ codes operated at the rate vector must tend to $1$ as $n$ grows. The proof is based on the method of types. The proof techniques are inspired by the work of Csiszár and Körner in 1982 which fully characterized the reliability function of any discrete memoryless channel (DMC) with feedback for rates above capacity. In addition, we generalize the strong converse theorem to the Gaussian model where each node is subject to a peak power constraint. Important consequences of our results are new strong converses for the Gaussian multiple access channel (MAC) with feedback and the following relay channels under both models: The degraded relay channel (RC), the RC with orthogonal sender components, and the general RC with feedback.
- In this paper, we analyze the asymptotics of the normalized remaining uncertainty of a source when a compressed or hashed version of it and correlated side-information is observed. For this system, commonly known as Slepian-Wolf source coding, we establish the optimal (minimum) rate of compression of the source to ensure that the remaining uncertainties vanish. We also study the exponential rate of decay of the remaining uncertainty to zero when the rate is above the optimal rate of compression. In our study, we consider various classes of random universal hash functions. Instead of measuring remaining uncertainties using traditional Shannon information measures, we do so using two forms of the conditional Rényi entropy. Among other techniques, we employ new one-shot bounds and the moments of type class enumerator method for these evaluations. We show that these asymptotic results are generalizations of the strong converse exponent and the error exponent of the Slepian-Wolf problem under maximum \empha posteriori (MAP) decoding.
- The objects of study of this paper are communication channels in which the dominant type of noise are symbol shifts, the main motivating examples being timing and bit-shift channels. Two channel models are introduced and their zero-error capacities and zero-error-detection capacities determined by explicit constructions of optimal codes. Model A can be informally described as follows: 1) The information is stored in an $ n $-cell register, where each cell is either empty or contains a particle of one of $ P $ possible types, and 2) due to the imperfections of the device each of the particles may be shifted several cells away from its original position over time. Model B is an abstraction of a single-server queue: 1) The transmitter sends packets from a $ P $-ary alphabet through a queuing system with an infinite buffer and a First-In-First-Out (FIFO) service procedure, and 2) each packet is being processed by the server for a random number of time slots. More general models including additional types of noise that the particles/packets can experience are also studied, as are the continuous-time versions of these problems.
- In this paper, a streaming transmission setup is considered where an encoder observes a new message in the beginning of each block and a decoder sequentially decodes each message after a delay of $T$ blocks. In this streaming setup, the fundamental interplay between the coding rate, the error probability, and the blocklength in the moderate deviations regime is studied. For output symmetric channels, the moderate deviations constant is shown to improve over the block coding or non-streaming setup by exactly a factor of $T$ for a certain range of moderate deviations scalings. For the converse proof, a more powerful decoder to which some extra information is fedforward is assumed. The error probability is bounded first for an auxiliary channel and this result is translated back to the original channel by using a newly developed change-of-measure lemma, where the speed of decay of the remainder term in the exponent is carefully characterized. For the achievability proof, a known coding technique that involves a joint encoding and decoding of fresh and past messages is applied with some manipulations in the error analysis.
- Motivated by streaming multi-view video coding and wireless sensor networks, we consider the problem of blockwise streaming compression of a pair of correlated sources, which we term streaming Slepian-Wolf coding. We study the moderate deviations regime in which the rate pairs of a sequence of codes converge, along a straight line, to various points on the boundary of the Slepian-Wolf region at a speed slower than the inverse square root of the blocklength $n$, while the error probability decays subexponentially fast in $n$. Our main result focuses on directions of approaches to corner points of the Slepian-Wolf region. It states that for each correlated source and all corner points, there exists a non-empty subset of directions of approaches such that the moderate deviations constant (the constant of proportionality for the subexponential decay of the error probability) is enhanced (over the non-streaming case) by at least a factor of $T$, the block delay of decoding source block pairs. We specialize our main result to the setting of streaming lossless source coding and generalize this result to the setting where we have different delay requirements for each of the two source blocks. The proof of our main result involves the use of various analytical tools and amalgamates several ideas from the recent information-theoretic streaming literature. We adapt the so-called truncated memory encoding idea from Draper and Khisti (2011) and Lee, Tan, and Khisti (2016) to ensure that the effect of error accumulation is nullified in the limit of large blocklengths. We also adapt the use of the so-called minimum weighted empirical suffix entropy decoder which was used by Draper, Chang, and Sahai (2014) to derive achievable error exponents for symbolwise streaming Slepian-Wolf coding.
- We propose a unified and systematic framework for performing online nonnegative matrix factorization in the presence of outliers. Our framework is particularly suited to large-scale data. We propose two solvers based on projected gradient descent and the alternating direction method of multipliers. We prove that the sequence of objective values converges almost surely by appealing to the quasi-martingale convergence theorem. We also show the sequence of learned dictionaries converges to the set of stationary points of the expected loss function almost surely. In addition, we extend our basic problem formulation to various settings with different constraints and regularizers. We also adapt the solvers and analyses to each setting. We perform extensive experiments on both synthetic and real datasets. These experiments demonstrate the computational efficiency and efficacy of our algorithms on tasks such as (parts-based) basis learning, image denoising, shadow removal and foreground-background separation.
- This paper revisits the Gaussian degraded relay channel, where the link that carries information from the source to the destination is a physically degraded version of the link that carries information from the source to the relay. The source and the relay are subject to expected power constraints. The $\varepsilon$-capacity of the channel is characterized and it is strictly larger than the capacity for $\varepsilon>0$, which implies that the channel does not possess the strong converse property. The proof of the achievability part is based on several key ideas: block Markov coding which is used in the classical decode-forward strategy, power control for Gaussian channels under expected power constraints, and a careful scaling between the block size and the total number of block uses. The converse part is proved by first establishing two non-asymptotic lower bounds on the error probability, which are derived from the type-II errors of some binary hypothesis tests. Subsequently, each lower bound is simplified by conditioning on an event related to the power of some linear combination of the codewords transmitted by the source and the relay. Lower and upper bounds on the second-order term of the optimal coding rate in terms of blocklength and error probability are also obtained.
- We study the top-$K$ ranking problem where the goal is to recover the set of top-$K$ ranked items out of a large collection of items based on partially revealed preferences. We consider an adversarial crowdsourced setting where there are two population sets, and pairwise comparison samples drawn from one of the populations follow the standard Bradley-Terry-Luce model (i.e., the chance of item $i$ beating item $j$ is proportional to the relative score of item $i$ to item $j$), while in the other population, the corresponding chance is inversely proportional to the relative score. When the relative size of the two populations is known, we characterize the minimax limit on the sample size required (up to a constant) for reliably identifying the top-$K$ items, and demonstrate how it scales with the relative size. Moreover, by leveraging a tensor decomposition method for disambiguating mixture distributions, we extend our result to the more realistic scenario in which the relative population size is unknown, thus establishing an upper bound on the fundamental limit of the sample size for recovering the top-$K$ set.
- We derive the optimal second-order coding region and moderate deviations constant for successive refinement source coding with a joint excess-distortion probability constraint. We consider two scenarios: (i) a discrete memoryless source (DMS) and arbitrary distortion measures at the decoders and (ii) a Gaussian memoryless source (GMS) and quadratic distortion measures at the decoders. For a DMS with arbitrary distortion measures, we prove an achievable second-order coding region, using type covering lemmas by Kanlis and Narayan and by No, Ingber and Weissman. We prove the converse using the perturbation approach by Gu and Effros. When the DMS is successively refinable, the expressions for the second-order coding region and the moderate deviations constant are simplified and easily computable. For this case, we also obtain new insights on the second-order behavior compared to the scenario where separate excess-distortion proabilities are considered. For example, we describe a DMS, for which the optimal second-order region transitions from being characterizable by a bivariate Gaussian to a univariate Gaussian, as the distortion levels are varied. We then consider a GMS with quadratic distortion measures. To prove the direct part, we make use of the sphere covering theorem by Verger-Gaugry, together with appropriately-defined Gaussian type classes. To prove the converse, we generalize Kostina and Verdú's one-shot converse bound for point-to-point lossy source coding. We remark that this proof is applicable to general successively refinable sources. In the proofs of the moderate deviations results for both scenarios, we follow a strategy similar to that for the second-order asymptotics and use the moderate deviations principle.
- This paper investigates the scaling exponent of polar codes for binary-input energy-harvesting (EH) channels with infinite-capacity batteries. The EH process is characterized by a sequence of i.i.d. random variables with finite variances. The scaling exponent $\mu$ of polar codes for a binary-input memoryless channel (BMC) characterizes the closest gap between the capacity and non-asymptotic rates achieved by polar codes with error probabilities no larger than some non-vanishing $\varepsilon\in(0,1)$. It has been shown that for any $\varepsilon\in(0,1)$, the scaling exponent $\mu$ for any binary-input memoryless symmetric channel (BMSC) with $I(q_{Y|X})\in(0,1)$ lies between 3.579 and 4.714 , where the upper bound $4.714$ was shown by an explicit construction of polar codes. Our main result shows that $4.714$ remains to be a valid upper bound on the scaling exponent for any binary-input EH channel, i.e., a BMC subject to additional EH constraints. Our result thus implies that the EH constraints do not worsen the rate of convergence to capacity if polar codes are employed. The main result is proved by leveraging the following three existing results: scaling exponent analyses for BMSCs, construction of polar codes designed for binary-input memoryless asymmetric channels, and the save-and-transmit strategy for EH channels.
- We consider streaming data transmission over a discrete memoryless channel. A new message is given to the encoder at the beginning of each block and the decoder decodes each message sequentially, after a delay of $T$ blocks. In this streaming setup, we study the fundamental interplay between the rate and error probability in the central limit and moderate deviations regimes and show that i) in the moderate deviations regime, the moderate deviations constant improves over the block coding or non-streaming setup by a factor of $T$ and ii) in the central limit regime, the second-order coding rate improves by a factor of approximately $\sqrt{T}$ for a wide range of channel parameters. For both regimes, we propose coding techniques that incorporate a joint encoding of fresh and previous messages. In particular, for the central limit regime, we propose a coding technique with truncated memory to ensure that a summation of constants, which arises as a result of applications of the central limit theorem, does not diverge in the error analysis. Furthermore, we explore interesting variants of the basic streaming setup in the moderate deviations regime. We first consider a scenario with an erasure option at the decoder and show that both the exponents of the total error and the undetected error probabilities improve by factors of $T$. Next, by utilizing the erasure option, we show that the exponent of the total error probability can be improved to that of the undetected error probability (in the order sense) at the expense of a variable decoding delay. Finally, we also extend our results to the case where the message rate is not fixed but alternates between two values.
- We study the second-order asymptotics of information transmission using random Gaussian codebooks and nearest neighbor (NN) decoding over a power-limited stationary memoryless additive non-Gaussian noise channel. We show that the dispersion term depends on the non-Gaussian noise only through its second and fourth moments, thus complementing the capacity result (Lapidoth, 1996), which depends only on the second moment. Furthermore, we characterize the second-order asymptotics of point-to-point codes over $K$-sender interference networks with non-Gaussian additive noise. Specifically, we assume that each user's codebook is Gaussian and that NN decoding is employed, i.e., that interference from the $K-1$ unintended users (Gaussian interfering signals) is treated as noise at each decoder. We show that while the first-order term in the asymptotic expansion of the maximum number of messages depends on the power of the interferring codewords only through their sum, this does not hold for the second-order term.
- In this paper, we consider single- and multi-user Gaussian channels with feedback under expected power constraints and with non-vanishing error probabilities. In the first of two contributions, we study asymptotic expansions for the additive white Gaussian noise (AWGN) channel with feedback under the average error probability formalism. By drawing ideas from Gallager and Nakiboğlu's work for the direct part and the meta-converse for the converse part, we establish the $\varepsilon$-capacity and show that it depends on $\varepsilon$ in general and so the strong converse fails to hold. Furthermore, we provide bounds on the second-order term in the asymptotic expansion. We show that for any positive integer $L$, the second-order term is bounded between a term proportional to $-\ln_{(L)} n$ (where $\ln_{(L)}(\cdot)$ is the $L$-fold nested logarithm function) and a term proportional to $+\sqrt{n\ln n}$ where $n$ is the blocklength. The lower bound on the second-order term shows that feedback does provide an improvement in the maximal achievable rate over the case where no feedback is available. In our second contribution, we establish the $\varepsilon$-capacity region for the AWGN multiple access channel (MAC) with feedback under the expected power constraint by combining ideas from hypothesis testing, information spectrum analysis, Ozarow's coding scheme, and power control.
- 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 (2015). 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 (2012). The converse part uses a perturbation argument by Gu and Effros (2009) in their strong converse proof of the discrete Gray-Wyner problem. The achievability part uses two novel elements: (i) a generalization of various type covering lemmas; and (ii) 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.
- This paper considers delay-limited communication over quasi-static fading channels under a long-term power constraint. A sequence of length-$n$ delay-limited codes for a quasi-static fading channel is said to be capacity-achieving if the codes achieve the delay-limited capacity, which is defined to be the maximum rate achievable by delay-limited codes. The delay-limited capacity is sometimes referred to as the zero-outage capacity in wireless communications. The delay-limited capacity is the appropriate choice of performance measure for delay-sensitive applications such as voice and video over fading channels. It is shown that for any sequence of capacity-achieving delay-limited codes with vanishing error probabilities, the normalized relative entropy between the output distribution induced by the length-$n$ code and the $n$-fold product of the capacity-achieving output distribution, denoted by $\frac{1}{n}D(p_{Y^n}\|p_{Y^n}^*)$, converges to zero. Additionally, we extend our convergence result to capacity-achieving delay-limited codes with non-vanishing error probabilities.
- We prove that the Gaussian broadcast channel with two destinations admits the strong converse property. This implies that for every sequence of block codes operated at a common rate pair with an asymptotic average error probability $<1$, the rate pair must lie within the capacity region derived by Cover and Bergmans. The main mathematical tool required for our analysis is a logarithmic Sobolev inequality known as the Gaussian Poincaré inequality.
- We derive upper and lower bounds for the error exponents of lossless streaming compression of two correlated sources under the blockwise and symbolwise settings. We consider the linear scaling regime in which the delay is a scalar multiple of the number of symbol pairs of interest. We show that for rate pairs satisfying certain constraints, the upper and lower bounds for the error exponent of blockwise codes coincide. For symbolwise codes, the bounds coincide for rate pairs satisfying the aforementioned constraints and a certain condition on the symbol pairs we wish to decode---namely, that their indices are asymptotically comparable to the blocklength. We also derive moderate deviations constants for blockwise and symbolwise codes, leveraging the error exponent results, and using appropriate Taylor series expansions. In particular, for blockwise codes, we derive an information spectrum-type strong converse, giving the complete characterization of the moderate deviations constants. For symbolwise codes, under an additional requirement on the backoff from the first-order fundamental limit, we can show that the moderate deviations constants are the same as the blockwise setting.
- This paper investigates the information-theoretic limits of energy-harvesting (EH) channels in the finite blocklength regime. The EH process is characterized by a sequence of i.i.d. random variables with finite variances. We use the save-and-transmit strategy proposed by Ozel and Ulukus (2012) together with Shannon's non-asymptotic achievability bound to obtain lower bounds on the achievable rates for both additive white Gaussian noise channels and discrete memoryless channels under EH constraints. The first-order terms of the lower bounds of the achievable rates are equal to $C$ and the second-order (backoff from capacity) terms are proportional to $-\sqrt{ \frac{\log n}{n}}$, where $n$ denotes the blocklength and $C$ denotes the capacity of the EH channel, which is the same as the capacity without the EH constraints. The constant of proportionality of the backoff term is found and qualitative interpretations are provided.
- In this paper, we study a security problem on a simple wiretap network, consisting of a source node S, a destination node D, and an intermediate node R. The intermediate node connects the source and the destination nodes via a set of noiseless parallel channels, with sizes $n_1$ and $n_2$, respectively. A message $M$ is to be sent from S to D. The information in the network may be eavesdropped by a set of wiretappers. The wiretappers cannot communicate with one another. Each wiretapper can access a subset of channels, called a wiretap set. All the chosen wiretap sets form a wiretap pattern. A random key $K$ is generated at S and a coding scheme on $(M, K)$ is employed to protect $M$. We define two decoding classes at D: In Class-I, only $M$ is required to be recovered and in Class-II, both $M$ and $K$ are required to be recovered. The objective is to minimize $H(K)/H(M)$ for a given wiretap pattern under the perfect secrecy constraint. The first question we address is whether routing is optimal on this simple network. By enumerating all the wiretap patterns on the Class-I/II $(3,3)$ networks and harnessing the power of Shannon-type inequalities, we find that gaps exist between the bounds implied by routing and the bounds implied by Shannon-type inequalities for a small fraction~($<2\%$) of all the wiretap patterns. The second question we investigate is the following: What is $\min H(K)/H(M)$ for the remaining wiretap patterns where gaps exist? We study some simple wiretap patterns and find that their Shannon bounds (i.e., the lower bound induced by Shannon-type inequalities) can be achieved by linear codes, which means routing is not sufficient even for the ($3$, $3$) network. For some complicated wiretap patterns, we study the structures of linear coding schemes under the assumption that they can achieve the corresponding Shannon bounds....
- We evaluate the asymptotics of equivocations, their exponents as well as their second-order coding rates under various Rényi information measures. Specifically, we consider the effect of applying a hash function on a source and we quantify the level of non-uniformity and dependence of the compressed source from another correlated source when the number of copies of the sources is large. Unlike previous works that use Shannon information measures to quantify randomness, information or uniformity, we define our security measures in terms of a more general class of information measures--the Rényi information measures and their Gallager-type counterparts. A special case of these Rényi information measure is the class of Shannon information measures. We prove tight asymptotic results for the security measures and their exponential rates of decay. We also prove bounds on the second-order asymptotics and show that these bounds match when the magnitudes of the second-order coding rates are large. We do so by establishing new classes non-asymptotic bounds on the equivocation and evaluating these bounds using various probabilistic limit theorems asymptotically.
- 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.
- We prove the strong converse for the $N$-source Gaussian multiple access channel (MAC). In particular, we show that any rate tuple that can be supported by a sequence of codes with asymptotic average error probability less than one must lie in the Cover-Wyner capacity region. Our proof consists of the following. First, we perform an expurgation step to convert any given sequence of codes with asymptotic average error probability less than one to codes with asymptotic maximal error probability less than one. Second, we quantize the input alphabets with an appropriately chosen resolution. Upon quantization, we apply the wringing technique (by Ahlswede) on the quantized inputs to obtain further subcodes from the subcodes obtained in the expurgation step so that the resultant correlations among the symbols transmitted by the different sources vanish as the blocklength grows. Finally, we derive upper bounds on achievable sum-rates of the subcodes in terms of the type-II error of a binary hypothesis test. These upper bounds are then simplified through judicious choices of auxiliary output distributions. Our strong converse result carries over to the Gaussian interference channel under strong interference as long as the sum of the two asymptotic average error probabilities less than one.
- Error and erasure exponents for the broadcast channel with degraded message sets are analyzed. The focus of our error probability analysis is on the main receiver where, nominally, both messages are to be decoded. A two-step decoding algorithm is proposed and analyzed. This receiver first attempts to decode both messages, failing which, it attempts to decode only the message representing the coarser information, i.e., the cloud center. This algorithm reflects the intuition that we should decode both messages only if we have confidence in the estimates; otherwise one should only decode the coarser information. The resulting error and erasure exponents, derived using the method of types, are expressed in terms of a penalized form of the modified random coding error exponent.
- This paper characterizes the second-order coding rates for lossy source coding with side information available at both the encoder and the decoder. We first provide non-asymptotic bounds for this problem and then specialize the non-asymptotic bounds for three different scenarios: discrete memoryless sources, Gaussian sources, and Markov sources. We obtain the second-order coding rates for these settings. It is interesting to observe that the second-order coding rate for Gaussian source coding with Gaussian side information available at both the encoder and the decoder is the same as that for Gaussian source coding without side information. Furthermore, regardless of the variance of the side information, the dispersion is $1/2$ nats squared per source symbol.
- This paper investigates the asymptotic expansion for the size of block codes defined for the additive white Gaussian noise (AWGN) channel with feedback under the following setting: A peak power constraint is imposed on every transmitted codeword, and the average error probability of decoding the transmitted message is non-vanishing as the blocklength increases. It is well-known that the presence of feedback does not increase the first-order asymptotics (i.e., capacity) in the asymptotic expansion for the AWGN channel. The main contribution of this paper is a self-contained proof of an upper bound on the asymptotic expansion for the AWGN channel with feedback. Combined with existing achievability results for the AWGN channel, our result implies that the presence of feedback does not improve the second- and third-order asymptotics. An auxiliary contribution is a proof of the strong converse for the parallel Gaussian channels with feedback under a peak power constraint.
- This paper studies the second-order asymptotics of the discrete memoryless multiple-access channel with degraded message sets. For a fixed average error probability $\epsilon\in(0,1)$ and an arbitrary point on the boundary of the capacity region, we characterize the speed of convergence of rate pairs that converge to that point for codes that have asymptotic error probability no larger than $\epsilon$, thus complementing an analogous result given previously for the Gaussian setting.
- This paper establishes that the strong converse holds for some classes of discrete memoryless multimessage multicast networks (DM-MMNs) whose corresponding cut-set bounds are tight, i.e., coincide with the set of achievable rate tuples. The strong converse for these classes of DM-MMNs implies that all sequences of codes with rate tuples belonging to the exterior of the cut-set bound have average error probabilities that necessarily tend to one (and are not simply bounded away from zero). Examples in the classes of DM-MMNs include wireless erasure networks, DM-MMNs consisting of independent discrete memoryless channels (DMCs) as well as single-destination DM-MMNs consisting of independent DMCs with destination feedback. Our elementary proof technique leverages properties of the Rényi divergence.
- The problem of channel coding with the erasure option is revisited for discrete memoryless channels. The interplay between the code rate, the undetected and total error probabilities is characterized. Using the information spectrum method, a sequence of codes of increasing blocklengths $n$ is designed to illustrate this tradeoff. Furthermore, for additive discrete memoryless channels with uniform input distribution, we establish that our analysis is tight with respect to the ensemble average. This is done by analysing the ensemble performance in terms of a tradeoff between the code rate, the undetected and the total errors. This tradeoff is parametrized by the threshold in a generalized likelihood ratio test. Two asymptotic regimes are studied. First, the code rate tends to the capacity of the channel at a rate slower than $n^{-1/2}$ corresponding to the moderate deviations regime. In this case, both error probabilities decay subexponentially and asymmetrically. The precise decay rates are characterized. Second, the code rate tends to capacity at a rate of $n^{-1/2}$. In this case, the total error probability is asymptotically a positive constant while the undetected error probability decays as $\exp(- b n^{ 1/2})$ for some $b>0$. The proof techniques involve applications of a modified (or "shifted") version of the Gärtner-Ellis theorem and the type class enumerator method to characterize the asymptotic behavior of a sequence of cumulant generating functions.
- We consider block codes for degraded wiretap channels in which the legitimate receiver decodes the message with an asymptotic error probability no larger than $\varepsilon$ but the leakage to the eavesdropper vanishes. For discrete memoryless and Gaussian wiretap channels, we show that the maximum rate of transmission does not depend on $\varepsilon\in [0,1)$, i.e., such channels possess the partial strong converse property. Furthermore, we derive sufficient conditions for the partial strong converse property to hold for memoryless but non-stationary symmetric and degraded wiretap channels. Our proof techniques leverage the information spectrum method, which allows us to establish a necessary and sufficient condition for the partial strong converse to hold for general wiretap channels without any information stability assumptions.
- We study a form of unequal error protection that we term "unequal message protection" (UMP). The message set of a UMP code is a union of $m$ disjoint message classes. Each class has its own error protection requirement, with some classes needing better error protection than others. We analyze the tradeoff between rates of message classes and the levels of error protection of these codes. We demonstrate that there is a clear performance loss compared to homogeneous (classical) codes with equivalent parameters. This is in sharp contrast to previous literature that considers UMP codes. To obtain our results we generalize finite block length achievability and converse bounds due to Polyanskiy-Poor-Verdú. We evaluate our bounds for the binary symmetric and binary erasure channels, and analyze the asymptotic characteristic of the bounds in the fixed error and moderate deviations regimes. In addition, we consider two questions related to the practical construction of UMP codes. First, we study a "header" construction that prefixes the message class into a header followed by data protection using a standard homogeneous code. We show that, in general, this construction is not optimal at finite block lengths. We further demonstrate that our main UMP achievability bound can be obtained using coset codes, which suggests a path to implementation of tractable UMP codes.
- In 1975, Carleial presented a special case of an interference channel in which the interference does not reduce the capacity of the constituent point-to-point Gaussian channels. In this work, we show that if the inequalities in the conditions that Carleial stated are strict, the dispersions are similarly unaffected. More precisely, in this work, we characterize the second-order coding rates of the Gaussian interference channel in the strictly very strong interference regime. In other words, we characterize the speed of convergence of rates of optimal block codes towards a boundary point of the (rectangular) capacity region. These second-order rates are expressed in terms of the average probability of error and variances of some modified information densities which coincide with the dispersion of the (single-user) Gaussian channel. We thus conclude that the dispersions are unaffected by interference in this channel model.
- We derive the optimum second-order coding rates, known as second-order capacities, for erasure and list decoding. For erasure decoding for discrete memoryless channels, we show that second-order capacity is $\sqrt{V}\Phi^{-1}(\epsilon_t)$ where $V$ is the channel dispersion and $\epsilon_t$ is the total error probability, i.e., the sum of the erasure and undetected errors. We show numerically that the expected rate at finite blocklength for erasures decoding can exceed the finite blocklength channel coding rate. We also show that the analogous result also holds for lossless source coding with decoder side information, i.e., Slepian-Wolf coding. For list decoding, we consider list codes of deterministic size that scales as $\exp(\sqrt{n}l)$ and show that the second-order capacity is $l+\sqrt{V}\Phi^{-1}(\epsilon)$ where $\epsilon$ is the permissible error probability. We also consider lists of polynomial size $n^\alpha$ and derive bounds on the third-order coding rate in terms of the order of the polynomial $\alpha$. These bounds are tight for symmetric and singular channels. The direct parts of the coding theorems leverage on the simple threshold decoder and converses are proved using variants of the hypothesis testing converse.
- This paper shows that, under the average error probability formalism, the third-order term in the normal approximation for the additive white Gaussian noise channel with a maximal or equal power constraint is at least $\frac{1}{2} \log n + O(1)$. This matches the upper bound derived by Polyanskiy-Poor-Verdú (2010).
- This paper studies the second-order asymptotics of the Gaussian multiple-access channel with degraded message sets. For a fixed average error probability $\varepsilon \in (0,1)$ and an arbitrary point on the boundary of the capacity region, we characterize the speed of convergence of rate pairs that converge to that boundary point for codes that have asymptotic error probability no larger than $\varepsilon$. As a stepping stone to this local notion of second-order asymptotics, we study a global notion, and establish relationships between the two. We provide a numerical example to illustrate how the angle of approach to a boundary point affects the second-order coding rate. This is the first conclusive characterization of the second-order asymptotics of a network information theory problem in which the capacity region is not a polygon.
- We study non-asymptotic fundamental limits for transmitting classical information over memoryless quantum channels, i.e. we investigate the amount of classical information that can be transmitted when a quantum channel is used a finite number of times and a fixed, non-vanishing average error is permissible. We consider the classical capacity of quantum channels that are image-additive, including all classical to quantum channels, as well as the product state capacity of arbitrary quantum channels. In both cases we show that the non-asymptotic fundamental limit admits a second-order approximation that illustrates the speed at which the rate of optimal codes converges to the Holevo capacity as the blocklength tends to infinity. The behavior is governed by a new channel parameter, called channel dispersion, for which we provide a geometrical interpretation.
- We study the performance limits of state-dependent discrete memoryless channels with a discrete state available at both the encoder and the decoder. We establish the epsilon-capacity as well as necessary and sufficient conditions for the strong converse property for such channels when the sequence of channel states is not necessarily stationary, memoryless or ergodic. We then seek a finer characterization of these capacities in terms of second-order coding rates. The general results are supplemented by several examples including i.i.d. and Markov states and mixed channels.
- Bounds on the reliability function for the discrete memoryless relay channel are derived using the method of types. Two achievable error exponents are derived based on partial decode-forward and compress-forward which are well-known superposition block-Markov coding schemes. The derivations require combinations of the techniques involved in the proofs of Csiszár-Körner-Marton's packing lemma for the error exponent of channel coding and Marton's type covering lemma for the error exponent of source coding with a fidelity criterion. The decode-forward error exponent is evaluated on Sato's relay channel. From this example, it is noted that to obtain the fastest possible decay in the error probability for a fixed effective coding rate, one ought to optimize the number of blocks in the block-Markov coding scheme assuming the blocklength within each block is large. An upper bound on the reliability function is also derived using ideas from Haroutunian's lower bound on the error probability for point-to-point channel coding with feedback.
- We present novel non-asymptotic or finite blocklength achievability bounds for three side-information problems in network information theory. These include (i) the Wyner-Ahlswede-Korner (WAK) problem of almost-lossless source coding with rate-limited side-information, (ii) the Wyner-Ziv (WZ) problem of lossy source coding with side-information at the decoder and (iii) the Gel'fand-Pinsker (GP) problem of channel coding with noncausal state information available at the encoder. The bounds are proved using ideas from channel simulation and channel resolvability. Our bounds for all three problems improve on all previous non-asymptotic bounds on the error probability of the WAK, WZ and GP problems--in particular those derived by Verdu. Using our novel non-asymptotic bounds, we recover the general formulas for the optimal rates of these side-information problems. Finally, we also present achievable second-order coding rates by applying the multidimensional Berry-Esseen theorem to our new non-asymptotic bounds. Numerical results show that the second-order coding rates obtained using our non-asymptotic achievability bounds are superior to those obtained using existing finite blocklength bounds.