arXiv subjects
Publications and source records attributed to Neri Merhav.
The output of a discrete Markov source is to be encoded instantaneously by a variable-rate encoder and decoded by a finite-state decoder. Our performance measure is a linear combination of the distortion and the instantaneous rate. Structure theorems, pertaining to the encoder and next-state functions are derived for every given finite-state decoder, which can have access to side information.
We propose a rigorous analysis approach for the subset sum problem in the context of lossless data compression, where the phase transition of the subset sum problem is directly related to the passage between ambiguous and non-ambiguous decompression, for a compression scheme that is based on specifying the sequence composition. The proposed analysis lends itself to straightforward extensions in several directions of interest, including non-binary alphabets, incorporation of side information at the decoder (Slepian-Wolf coding), and coding schemes based on multiple subset sums. It is also demonstrated that the proposed technique can be used to analyze the critical behavior in a more involved situation where the sequence composition is not specified by the encoder.
The average redundancy of the Shannon code, $R_n$, as a function of the block length $n$, is known to exhibit two very different types of behavior, depending on the rationality or irrationality of certain parameters of the source: It either converges to 1/2 as $n$ grows without bound, or it may have a non-vanishing, oscillatory, (quasi-) periodic pattern around the value 1/2 for all large $n$. In this paper, we make an attempt to shed some insight into this erratic behavior of $R_n$, by drawing an analogy with the realm of physics of wave propagation, in particular, the elementary theory of scattering and diffraction. It turns out that there are two types of behavior of wave diffraction patterns formed by crystals, which are correspondingly analogous to the two types of patterns of $R_n$. When the crystal is perfect, the diffraction intensity spectrum exhibits very sharp peaks, a.k.a. Bragg peaks, at wavelengths of full constructive interference. These wavelengths correspond to the frequencies of the harmonic waves of the oscillatory mode of $R_n$. On the other hand, when the crystal is imperfect and there is a considerable degree of disorder in its structure, the Bragg peaks disappear, and the behavior of this mode is analogous to the one where $R_n$ is convergent.
We consider a general problem of finding a strategy that minimizes the exponential moment of a given cost function, with an emphasis on its relation to the more common criterion of minimization the expectation of the first moment of the same cost function. In particular, our main result is a theorem that gives simple sufficient conditions for a strategy to be optimum in the exponential moment sense. This theorem may be useful in various situations, and application examples are given. We also examine the asymptotic regime and investigate universal asymptotically optimum strategies in light of the aforementioned sufficient conditions, as well as phenomena of irregularities, or phase transitions, in the behavior of the asymptotic performance, which can be viewed and understood from a statistical-mechanical perspective. Finally, we propose a new route for deriving lower bounds on exponential moments of certain cost functions (like the square error in estimation problems) on the basis of well known lower bounds on their expectations.
We draw a certain analogy between the classical information-theoretic problem of lossy data compression (source coding) of memoryless information sources and the statistical mechanical behavior of a certain model of a chain of connected particles (e.g., a polymer) that is subjected to a contracting force. The free energy difference pertaining to such a contraction turns out to be proportional to the rate-distortion function in the analogous data compression model, and the contracting force is proportional to the derivative this function. Beyond the fact that this analogy may be interesting on its own right, it may provide a physical perspective on the behavior of optimum schemes for lossy data compression (and perhaps also, an information-theoretic perspective on certain physical system models). Moreover, it triggers the derivation of lossy compression performance for systems with memory, using analysis tools and insights from statistical mechanics.
Several implications of well-known fluctuation theorems, on the statistical properties of the entropy production, are studied using various approaches. We begin by deriving a tight lower bound on the variance of the entropy production for a given mean of this random variable. It is shown that the Evans-Searles fluctuation theorem alone imposes a significant lower bound on the variance only when the mean entropy production is very small. It is then nonetheless demonstrated that upon incorporating additional information concerning the entropy production, this lower bound can be significantly improved, so as to capture extensivity properties. Another important aspect of the fluctuation properties of the entropy production is the relationship between the mean and the variance, on the one hand, and the probability of the event where the entropy production is negative, on the other hand. Accordingly, we derive upper and lower bounds on this probability in terms of the mean and the variance. These bounds are tighter than previous bounds that can be found in the literature. Moreover, they are tight in the sense that there exist probability distributions, satisfying the Evans-Searles fluctuation theorem, that achieve them with equality. Finally, we present a general method for generating a wide class of inequalities that must be satisfied by the entropy production. We use this method to derive several new inequalities which go beyond the standard derivation of the second law.
We draw relationships between the generalized data processing theorems of Zakai and Ziv (1973 and 1975) and the dynamical version of the second law of thermodynamics, a.k.a. the Boltzmann H-Theorem, which asserts that the Shannon entropy, $H(X_t)$, pertaining to a finite--state Markov process $\{X_t\}$, is monotonically non-decreasing as a function of time $t$, provided that the steady-state distribution of this process is uniform across the state space (which is the case when the process designates an isolated system). It turns out that both the generalized data processing theorems and the Boltzmann H-Theorem can be viewed as special cases of a more general principle concerning the monotonicity (in time) of a certain generalized information measure applied to a Markov process. This gives rise to a new look at the generalized data processing theorem, which suggests to exploit certain degrees of freedom that may lead to better bounds, for a given choice of the convex function that defines the generalized mutual information.
This document consists of lecture notes for a graduate course, which focuses on the relations between Information Theory and Statistical Physics. The course is aimed at EE graduate students in the area of Communications and Information Theory, as well as to graduate students in Physics who have basic background in Information Theory. Strong emphasis is given to the analogy and parallelism between Information Theory and Statistical Physics, as well as to the insights, the analysis tools and techniques that can be borrowed from Statistical Physics and `imported' to certain problem areas in Information Theory. This is a research trend that has been very active in the last few decades, and the hope is that by exposing the student to the meeting points between these two disciplines, we will enhance his/her background and perspective to carry out research in the field. A short outline of the course is as follows: Introduction; Elementary Statistical Physics and its Relation to Information Theory; Analysis Tools in Statistical Physics; Systems of Interacting Particles and Phase Transitions; The Random Energy Model (REM) and Random Channel Coding; Additional Topics (optional).
Threshold effects in the estimation of parameters of non-linearly modulated, continuous-time, wide-band waveforms, are examined from a statistical physics perspective. These threshold effects are shown to be analogous to phase transitions of certain disordered physical systems in thermal equilibrium. The main message, in this work, is in demonstrating that this physical point of view may be insightful for understanding the interactions between two or more parameters to be estimated, from the aspects of the threshold effect.
We derive a simple general parametric representation of the rate-distortion function of a memoryless source, where both the rate and the distortion are given by integrals whose integrands include the minimum mean square error (MMSE) of the distortion $Δ=d(X,Y)$ based on the source symbol $X$, with respect to a certain joint distribution of these two random variables. At first glance, these relations may seem somewhat similar to the I-MMSE relations due to Guo, Shamai and Verdú, but they are, in fact, quite different. The new relations among rate, distortion, and MMSE are discussed from several aspects, and more importantly, it is demonstrated that they can sometimes be rather useful for obtaining non-trivial upper and lower bounds on the rate-distortion function, as well as for determining the exact asymptotic behavior for very low and for very large distortion. Analogous MMSE relations hold for channel capacity as well.
We study the large deviations behavior of systems that admit a certain form of a product distribution, which is frequently encountered both in Physics and in various information system models. First, to fix ideas, we demonstrate a simple calculation of the large deviations rate function for a single constraint (event). Under certain conditions, the behavior of this function is shown to exhibit an analogue of Bose--Einstein condensation (BEC). More interestingly, we also study the large deviations rate function associated with two constraints (and the extension to any number of constraints is conceptually straightforward). The phase diagram of this rate function is shown to exhibit as many as seven phases, and it suggests a two--dimensional generalization of the notion of BEC (or more generally, a multi--dimensional BEC). While the results are illustrated for a simple model, the underlying principles are actually rather general. We also discuss several applications and implications pertaining to information system models.
In continuation to a recent work on the statistical--mechanical analysis of minimum mean square error (MMSE) estimation in Gaussian noise via its relation to the mutual information (the I-MMSE relation), here we propose a simple and more direct relationship between optimum estimation and certain information measures (e.g., the information density and the Fisher information), which can be viewed as partition functions and hence are amenable to analysis using statistical--mechanical techniques. The proposed approach has several advantages, most notably, its applicability to general sources and channels, as opposed to the I-MMSE relation and its variants which hold only for certain classes of channels (e.g., additive white Gaussian noise channels). We then demonstrate the derivation of the conditional mean estimator and the MMSE in a few examples. Two of these examples turn out to be generalizable to a fairly wide class of sources and channels. For this class, the proposed approach is shown to yield an approximate conditional mean estimator and an MMSE formula that has the flavor of a single-letter expression. We also show how our approach can easily be generalized to situations of mismatched estimation.
We revisit and extend the physical interpretation recently given to a certain identity between large--deviations rate--functions (as well as applications of this identity to Information Theory), as an instance of thermal equilibrium between several physical systems that are brought into contact. Our new interpretation, of mechanical equilibrium between these systems, is shown to have several advantages relative to that of thermal equilibrium. This physical point of view also provides a trigger to the development of certain alternative representations of the rate--distortion function and channel capacity, which are new to the best knowledge of the author.
We present adaptive on-line schemes for lossy encoding of individual sequences under the conditions of the Wyner-Ziv (WZ) problem. In the first part of this article, a set of fixed-rate scalar source codes with zero delay is presented. We propose a randomized on-line coding scheme, which achieves asymptotically (and with high probability), the performance of the best source code in the set, uniformly over all source sequences. The scheme uses the same rate and has zero delay. We then present an efficient algorithm for implementing our on-line coding scheme in the case of a relatively small set of encoders. We also present an efficient algorithm for the case of a larger set of encoders with a structure, using the method of the weighted graph and the Weight Pushing Algorithm (WPA). In the second part of this article, we extend our results to the case of variable-rate coding. A set of variable-rate scalar source codes is presented. We generalize the randomized on-line coding scheme, to our case. This time, the performance is measured by the Lagrangian Cost (LC), which is defined as a weighted sum of the distortion and the length of the encoded sequence. We present an efficient algorithm for implementing our on-line variable-rate coding scheme in the case of a relatively small set of encoders. We then consider the special case of lossless variable-rate coding. An on-line scheme which use Huffman codes is presented. We show that this scheme can be implemented efficiently using the same graphic methods from the first part. Combining the results from former sections, we build a generalized efficient algorithm for structured set of variable-rate encoders. The complexity of all the algorithms is no more than linear in the sequence length.
We consider a broadcast channel with a degraded message set, in which a single transmitter sends a common message to two receivers and a private message to one of the receivers only. The main goal of this work is to find new lower bounds to the error exponents of the strong user, the one that should decode both messages, and of the weak user, that should decode only the common message. Unlike previous works, where suboptimal decoders where used, the exponents we derive in this work pertain to optimal decoding and depend on both rates. We take two different approaches. The first approach is based, in part, on variations of Gallager-type bounding techniques that were presented in a much earlier work on error exponents for erasure/list decoding. The resulting lower bounds are quite simple to understand and to compute. The second approach is based on a technique that is rooted in statistical physics, and it is exponentially tight from the initial step and onward. This technique is based on analyzing the statistics of certain enumerators. Numerical results show that the bounds obtained by this technique are tighter than those obtained by the first approach and previous results. The derivation, however, is more complex than the first approach and the retrieved exponents are harder to compute.
We provide a simple physical interpretation, in the context of the second law of thermodynamics, to the information inequality (a.k.a. the Gibbs' inequality, which is also equivalent to the log-sum inequality), asserting that the relative entropy between two probability distributions cannot be negative. Since this inequality stands at the basis of the data processing theorem (DPT), and the DPT in turn is at the heart of most, if not all, proofs of converse theorems in Shannon theory, it is observed that conceptually, the roots of fundamental limits of Information Theory can actually be attributed to the laws of physics, in particular, to the second law of thermodynamics, and at least indirectly, also to the law of energy conservation. By the same token, in the other direction: one can view the second law as stemming from information-theoretic principles.
We consider a decoder with an erasure option and a variable size list decoder for channels with non-casual side information at the transmitter. First, universally achievable error exponents are offered for decoding with an erasure option using a parameterized decoder in the spirit of Csiszár and Körner's decoder. Then, the proposed decoding rule is generalized by extending the range of its parameters to allow variable size list decoding. This extension gives a unified treatment for erasure/list decoding. Exponential bounds on the probability of list error and the average number of incorrect messages on the list are given. Relations to Forney's and Csiszár and Körner's decoders for discrete memoryless channel are discussed. These results are obtained by exploring a random binning code with conditionally constant composition codewords proposed by Moulin and Wang, but with a different decoding rule.
We consider the problem of signal estimation (denoising) from a statistical mechanical perspective, using a relationship between the minimum mean square error (MMSE), of estimating a signal, and the mutual information between this signal and its noisy version. The paper consists of essentially two parts. In the first, we derive several statistical-mechanical relationships between a few important quantities in this problem area, such as the MMSE, the differential entropy, the Fisher information, the free energy, and a generalized notion of temperature. We also draw analogies and differences between certain relations pertaining to the estimation problem and the parallel relations in thermodynamics and statistical physics. In the second part of the paper, we provide several application examples, where we demonstrate how certain analysis tools that are customary in statistical physics, prove useful in the analysis of the MMSE. In most of these examples, the corresponding statistical-mechanical systems turn out to consist of strong interactions that cause phase transitions, which in turn are reflected as irregularities and discontinuities (similar to threshold effects) in the behavior of the MMSE.