This paper explores a predictive game in which a Forecaster announces odds based on a time-homogeneous Markov kernel, establishing a game-theoretic law of large numbers for the relative frequencies of occurrences of all finite strings. A key feature of our proof is a betting strategy built on a universal coding scheme, inspired by the martingale convergence theorem and algorithmic randomness theory, without relying on a diversified betting approach that involves countably many operating accounts. We apply these insights to thermodynamics, offering a game-theoretic perspective on Leó Szilárd's thought experiment.
We derive novel upper and lower finite-length bounds of the error probability in joint source-channel coding when the source obeys an ergodic Markov process and the channel is a Markovian additive channel or a Markovian conditional additive channel. These bounds are tight in the large and moderate deviation regimes.
One open problem in source coding is to characterize the limits of representing losslessly a non-identity discrete function of the data encoded independently by the encoders of several correlated sources with memory. This paper investigates this problem under Markovian conditions, namely either the sources or the functions considered are Markovian. We propose using linear mappings over finite rings as encoders. If the function considered admits certain polynomial structure, the linear encoders can make use of this structure to establish "implicit collaboration" and boost the performance. In fa…
We address the problem of simulating an arbitrary Markovian interactive protocol over binary symmetric channels with crossover probability $\varepsilon$. We are interested in the achievable rates of reliable simulation, i.e., in characterizing the smallest possible blowup in communications such that a vanishing error probability (in the protocol length) can be attained. Whereas for general interactive protocols the output of each party may depend on all previous outputs of its counterpart, in a (first order) Markovian protocol this dependence is limited to the last observed output only. In the…
In this work, we propose solving the Information bottleneck (IB) and Privacy Funnel (PF) problems with Douglas-Rachford Splitting methods (DRS). We study a general Markovian information-theoretic Lagrangian that includes IB and PF into a unified framework. We prove the linear convergence of the proposed solvers using the Kurdyka-Łojasiewicz inequality. Moreover, our analysis is beyond IB and PF and applies to any convex-weakly convex pair objectives. Based on the results, we develop two types of linearly convergent IB solvers, with one improves the performance of convergence over existing solv…
In this paper, we study the asymptotic performance of Abelian group codes for the lossy source coding problem for arbitrary discrete (finite alphabet) memoryless sources as well as the channel coding problem for arbitrary discrete (finite alphabet) memoryless channels. For the source coding problem, we derive an achievable rate-distortion function that is characterized in a single-letter information-theoretic form using the ensemble of Abelian group codes. When the underlying group is a field, it simplifies to the symmetric rate-distortion function. Similarly, for the channel coding problem, w…
Consider a communication scenario over a noiseless channel where a sender is required to broadcast messages to multiple receivers, each having side information about some messages. In this scenario, the sender can leverage the receivers' side information during the encoding of messages in order to reduce the required transmissions. This type of encoding is called index coding. In this paper, we study index coding with two cooperative senders, each with some subset of messages, and multiple receivers, each requesting one unique message. The index coding in this setup is called two-sender unicas…
Information-theoretic work for wiretap channels is mostly based on random coding schemes. Designing practical coding schemes to achieve information-theoretic security is an important problem. By applying the two recently developed techniques for polar codes, we propose a polar coding scheme to achieve the secrecy capacity of the general wiretap channel.
The error exponent of a discrete memoryless channel is expressed in two forms. One is Gallager's expression with a positive slope parameter and the other is Csiszar and Korner's information-theoretic representation expressed using the mutual information and the relative entropy. They differ in appearance, and existing methods to prove their agreement are not elementary, as they require an evaluation of the KKT conditions that the optimal distribution must satisfy. Similarly, there are two types of expressions for the strong converse exponent. They are Arimoto's expression with a negative slope…
We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic factor from the chromatic number. This is in striking contrast to minrank since prior work has shown that it can outperform the chromatic number by a polynomial factor in some cases. The conclusion is that all known graph theoretic bounds are not much stronger than the chromatic number.
We consider linear network error correction (LNEC) coding when errors may occur on edges of a communication network of which the topology is known. In this paper, we first revisit and explore the framework of LNEC coding, and then unify two well-known LNEC coding approaches. Furthermore, by developing a graph-theoretic approach to the framework of LNEC coding, we obtain a significantly enhanced characterization of the error correction capability of LNEC codes in terms of the minimum distances at the sink nodes. In LNEC coding, the minimum required field size for the existence of LNEC codes, in…
Baranyai's theorem is a well-known theorem in the theory of hypergraphs. A corollary of this theorem says that one can partition the family of all $u$-subsets of an $n$-element set into ${n-1\choose u-1}$ sub-families such that each sub-family form a partition of the $n$-element set, where $n$ is divisible by $u$. In this paper, we present a coding-theoretic application of Baranyai's theorem (or equivalently, the corollary). More precisely, we propose the first purely combinatorial construction of locally decodable codes. Locally decodable codes are error-correcting codes that allow the recove…
We consider a communication network where there exist wiretappers who can access a subset of channels, called a wiretap set, which is chosen from a given collection of wiretap sets. The collection of wiretap sets can be arbitrary. Secure network coding is applied to prevent the source information from being leaked to the wiretappers. In secure network coding, the required alphabet size is an open problem not only of theoretical interest but also of practical importance, because it is closely related to the implementation of such coding schemes in terms of computational complexity and storage r…
A common assumption in the existing network coding literature is that the users are cooperative and non-selfish. However, this assumption can be violated in practice. In this paper, we analyze inter-session network coding in a wired network using game theory. We assume selfish users acting strategically to maximize their own utility, leading to a resource allocation game among users. In particular, we study the well-known butterfly network topology where a bottleneck link is shared by several network coding and routing flows. We prove the existence of a Nash equilibrium for a wide range of uti…
In applications of distributed storage systems to distributed computing and implementation of key- value stores, the following property, usually referred to as consistency in computer science and engineering, is an important requirement: as the data stored changes, the latest version of the data must be accessible to a client that connects to the storage system. An information theoretic formulation called multi-version coding is introduced in the paper, in order to study storage costs of consistent distributed storage systems. Multi-version coding is characterized by ν totally ordered versions…
Under the paradigm of caching, partial data is delivered before the actual requests of users are known. In this paper, this problem is modeled as a canonical distributed source coding problem with side information, where the side information represents the users' requests. For the single-user case, a single-letter characterization of the optimal rate region is established, and for several important special cases, closed-form solutions are given, including the scenario of uniformly distributed user requests. In this case, it is shown that the optimal caching strategy is closely related to total…
We review connections between coding-theoretic objects and sparse learning problems. In particular, we show how seemingly different combinatorial objects such as error-correcting codes, combinatorial designs, spherical codes, compressed sensing matrices and group testing designs can be obtained from one another. The reductions enable one to translate upper and lower bounds on the parameters attainable by one object to another. We survey some of the well-known reductions in a unified presentation, and bring some existing gaps to attention. New reductions are also introduced; in particular, we b…
This paper provides a source coding theorem for multi-dimensional information signals when, at a given instant, the distribution associated with one arbitrary component of the signal to be compressed is not known and a side information is available at the destination. This new framework appears to be both of information-theoretical and game-theoretical interest: it provides a new type of constraints to compress an information source; it is useful for designing certain types of mediators in games and characterize utility regions for games with signals. Regarding the latter aspect, we apply the …
The problem of serving multicast flows in a crossbar switch is considered. Intra-flow linear network coding is shown to achieve a larger rate region than the case without coding. A traffic pattern is presented which is achievable with coding but requires a switch speedup when coding is not allowed. The rate region with coding can be characterized in a simple graph-theoretic manner, in terms of the stable set polytope of the "enhanced conflict graph". No such graph-theoretic characterization is known for the case of fanout splitting without coding. The minimum speedup needed to achieve 100% thr…
We show a construction of a quantum ramp secret sharing scheme from a nested pair of linear codes. Necessary and sufficient conditions for qualified sets and forbidden sets are given in terms of combinatorial properties of nested linear codes. An algebraic geometric construction for quantum secret sharing is also given.
Johns Hopkins University School of Medicine（约翰·霍普金斯大学医学院）；China Electronics Technology Group Corporation（中国电子科技集团公司）
While Shannon's mutual information has widespread applications in many disciplines, for practical applications it is often difficult to calculate its value accurately for high-dimensional variables because of the curse of dimensionality. This paper is focused on effective approximation methods for evaluating mutual information in the context of neural population coding. For large but finite neural populations, we derive several information-theoretic asymptotic bounds and approximation formulas that remain valid in high-dimensional spaces. We prove that optimizing the population density distrib…
We study the information leakage to a guessing adversary in zero-error source coding. The source coding problem is defined by a confusion graph capturing the distinguishability between source symbols. The information leakage is measured by the ratio of the adversary's successful guessing probability after and before eavesdropping the codeword, maximized over all possible source distributions. Such measurement under the basic adversarial model where the adversary makes a single guess and allows no distortion between its estimator and the true sequence is known as the maximum min-entropy leakage…
A communication link aided by a reconfigurable intelligent surface (RIS) is studied in which the transmitter can control the state of the RIS via a finite-rate control link. Channel state information (CSI) is acquired at the receiver based on pilot-assisted channel estimation, and it may or may not be shared with the transmitter. Considering quasi-static fading channels with imperfect CSI, capacity-achieving signalling is shown to implement joint encoding of the transmitted signal and of the response of the RIS. This demonstrates the information-theoretic optimality of RIS-based modulation, or…
