Exponent Of Probability Research Articles

Decoding error probability in the erasure channel and the $\boldsymbol~r$-th support weight distribution

Inspired by a recent work of Shen and Fu (2019), we study the decoding error probability of linear codes over the erasure channel under three decoding principles. We first simplify the explicit formulas obtained by Shen and Fu (2019), showing a much more direct relation to the $r$-th support weight distribution of linear codes. Then we compute the average $r$-th support weight distribution of $[n,k]_q$ linear codes. We resolve an open question of Shen and Fu (2019) concerning the error exponent of the average unsuccessful decoding probability under maximum likelihood decoding for random $[n,nR]_q$ linear codes. We also obtain the error exponent of the average unsuccessful decoding probability for random $[n,nR]_q$ linear codes under list decoding.

Asymptotically Optimal One- and Two-Sample Testing With Kernels

We characterize the asymptotic performance of nonparametric one- and two-sample testing. The exponential decay rate or error exponent of the type-II error probability is used as the asymptotic performance metric, and an optimal test achieves the maximum rate subject to a constant level constraint on the type-I error probability. With Sanov's theorem, we derive a sufficient condition for one-sample tests to achieve the optimal error exponent in the universal setting, i.e., for any distribution defining the alternative hypothesis. We then show that two classes of Maximum Mean Discrepancy (MMD) based tests attain the optimal type-II error exponent on \mathbb R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> , while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve this optimality with an asymptotic level constraint. For general two-sample testing, however, Sanov's theorem is insufficient to obtain a similar sufficient condition. We proceed to establish an extended version of Sanov's theorem and derive an exact error exponent for the quadratic-time MMD based two-sample tests. The obtained error exponent is further shown to be optimal among all two-sample tests satisfying a given level constraint. Our work hence provides an achievability result for optimal nonparametric one- and two-sample testing in the universal setting. Application to off-line change detection and related issues are also discussed.

Privacy-Utility Tradeoff for Hypothesis Testing Over a Noisy Channel

We study a hypothesis testing problem with a privacy constraint over a noisy channel and derive the performance of optimal tests under the Neyman-Pearson criterion. The fundamental limit of interest is the privacy-utility tradeoff (PUT) between the exponent of the type-II error probability and the leakage of the information source subject to a constant constraint on the type-I error probability. We provide an exact characterization of the asymptotic PUT for any non-vanishing type-I error probability. Our result implies that tolerating a larger type-I error probability cannot improve the PUT. Such a result is known as a strong converse or strong impossibility theorem. To prove the strong converse theorem, we apply the recently proposed technique in (Tyagi and Watanabe, 2020) and further demonstrate its generality. The strong converse theorems for several problems, such as hypothesis testing against independence over a noisy channel (Sreekumar and Gündüz, 2020) and hypothesis testing with communication and privacy constraints (Gilani et al., 2020), are established or recovered as special cases of our result.

Bifractal nature of chromosome contact maps

Modern biological techniques such as Hi-C permit to measure probabilities that different chromosomal regions are close in space. These probabilities can be visualised as matrices called contact maps. In this paper, we introduce a multifractal analysis of chromosomal contact maps. Our analysis reveals that Hi-C maps are bifractal, i.e. complex geometrical objects characterized by two distinct fractal dimensions. To rationalize this observation, we introduce a model that describes chromosomes as a hierarchical set of nested domains and we solve it exactly. The predicted multifractal spectrum is in excellent quantitative agreement with experimental data. Moreover, we show that our theory yields to a more robust estimation of the scaling exponent of the contact probability than existing methods. By applying this method to experimental data, we detect subtle conformational changes among chromosomes during differentiation of human stem cells.

Empirical analysis on human dynamics of sharing-bicycles’ user behavior

With the development of mobile communication and global positioning system navigation and positioning technology, analysis of user behavior on mobile Internet has become a hot topic in research area. Sign in sharing-bicycles’ app, find bicycle location, and selected has become a part of mobile Internet user’s daily life. Based on the data analysis of the spatial and temporal characteristics, find out sharing-bicycle’s user behavior obeys power-law distribution. In the time interval, user behavior of sharing-bicycle has strong intermittency and weak memory; the exponent of probability that K edge nodes is three by fitting the distance of sharing-bicycle’s data curve. It is verified that mobile Internet is long to scale-free networks. We conclude seven characteristics of user’s behavior of sharing-bicycle in mobile Internet application from experimental results. The analysis of sharing-bicycle’s behavior has become a complement and extension in human dynamics research field.

Anonymous Heterogeneous Distributed Detection: Optimal Decision Rules, Error Exponents, and the Price of Anonymity

We explore the fundamental limits of heterogeneous distributed detection in an anonymous sensor network with $n$ sensors and a single fusion center. The fusion center collects the single observation from each of the $n$ sensors to detect a binary parameter. The sensors are clustered into multiple groups, and different groups follow different distributions under a given hypothesis. The key challenge for the fusion center is the anonymity of sensors—although it knows the exact number of sensors and the distribution of observations in each group, it does not know which group each sensor belongs to. It is hence natural to consider it as a composite hypothesis testing problem. First, we propose an optimal test called mixture likelihood ratio test , which is a randomized threshold test based on the ratio of the uniform mixture of all the possible distributions under one hypothesis to that under the other hypothesis. Optimality is shown by first arguing that there exists an optimal test that is symmetric , that is, it does not depend on the order of observations across the sensors, and then proving that the mixture likelihood ratio test is optimal among all symmetric tests. Second, we focus on the Neyman–Pearson setting and characterize the error exponent of the worst-case type-II error probability as $n$ tends to infinity, assuming the number of sensors in each group is proportional to $n$ . Finally, we generalize our result to find the collection of all achievable type-I and type-II error exponents, showing that the boundary of the region can be obtained by solving an optimization problem. Our results elucidate the price of anonymity in heterogeneous distributed detection, and can be extended to $M$ -ary hypothesis testing with heterogeneous observations generated according to hidden latent variables.

The Decoding Error Probability of Linear Codes Over the Erasure Channel

In this paper, we study the decoding error probability of linear codes over the erasure channel under the list decoding. The notion of the $q^\ell $ -incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding. By calculating the $q^\ell $ -incorrigible set distributions, the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions. For the ensemble of all $[n,k]$ linear codes, the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined. Furthermore, the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all $[n,nR]$ linear codes.

Streaming Data Transmission in the Moderate Deviations and Central Limit Regimes

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: 1) in the moderate deviations regime, the moderate deviations constant improves over the block coding or non-streaming setup by a factor of $T$ and 2) 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 the 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, i.e., the decoder can output an erasure symbol instead of a message estimate, 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.

Secret Key-Based Identification and Authentication With a Privacy Constraint

We consider the problem of identification and authentication based on secret key generation from some user-generated source data (e.g., a biometric source). The goal is to reliably identify users pre-enrolled in a database as well as authenticate them based on the estimated secret key while preserving the privacy of the enrolled data and of the generated keys. We characterize the optimal tradeoff region of the identification rate, compression rate of the users’ source data, information leakage rate, and secret key rate. In particular, we provide a coding strategy based on layered random binning which is shown to be optimal. In addition, we study a related secure identification/authentication problem where an adversary tries to deceive the system using its own data. Here, the optimal tradeoff of the identification rate, compression rate, leakage rate, and exponent of the maximum false acceptance probability is provided. The results reveal a close connection between the optimal secret key rate and the false acceptance exponent of the identification/authentication system.

Tangled nature model of evolutionary dynamics reconsidered: Structural and dynamical effects of trait inheritance

Based on the stochastic dynamics of interacting agents which reproduce, mutate, and die, the tangled nature model (TNM) describes key emergent features of biological and cultural ecosystems' evolution. While trait inheritance is not included in many applications, i.e., the interactions of an agent and those of its mutated offspring are taken to be uncorrelated, in the family of TNMs introduced in this work correlations of varying strength are parametrized by a positive integer K. We first show that the interactions generated by our rule are nearly independent of K. Consequently, the structural and dynamical effects of trait inheritance can be studied independently of effects related to the form of the interactions. We then show that changing K strengthens the core structure of the ecology, leads to population abundance distributions better approximated by log-normal probability densities, and increases the probability that a species extant at time t_{w} also survives at t>t_{w}. Finally, survival probabilities of species are shown to decay as powers of the ratio t/t_{w}, a so-called pure aging behavior usually seen in glassy systems of physical origin. We find a quantitative dynamical effect of trait inheritance, namely, that increasing the value of K numerically decreases the decay exponent of the species survival probability.

A Modified Energy Detector for Random Signals in Gaussian Noise

This letter proposes a new modified energy detector for random signals in Gaussian noise. This new detector, whose decision variable is the weighted sum of the signal amplitude and its square, has a simple structure similar to the conventional energy detector. We propose a parameter optimization method to find the proper weighting factor by maximizing the error exponent of the misdetection probability based on large deviation theory. We also apply this method to optimize the power operation $p$ of another modified energy detector recently proposed by Chen. Numerical results show that, with the parameters obtained by the proposed method, both modified detectors achieve performance almost as well as the optimal detector.

Error Exponent for Multiple Access Channels: Upper Bounds

The problem of bounding the reliability function of a multiple access channel (MAC) is studied. Two new upper bounds on the error exponent of a two-user discrete memoryless (DM)-MAC are derived. The first bound (sphere packing) is an upper bound on the exponent of the average probability of error and is the first bound of this type that is zero outside the capacity region and thus results in a tighter sphere-packing exponent when compared with the tightest known exponent derived by Haroutunian. The second bound (minimum distance) is an upper bound on the exponent of the maximal (as opposed to average) probability of error. To obtain this bound, first, an upper bound on the minimum Bhattacharyya distance between codeword pairs is derived. For a certain class of two-user DM-MACs, an upper bound on the exponent of maximal probability of error is derived as a consequence of the upper bound on the minimum Bhattacharyya distance. We analytically evaluate the sphere packing bound for uniform composition codes for an additive and nonsymmetric channel and show that it is tight near the boundary of the capacity region, i.e., equal to the random coding lower bound.

Least action principle and stochastic motion : a generic derivation of path probability

This work is an analytical calculation of the path probability for random dynamics of mechanical system described by Langevin equation with Gaussian noise. The result shows an exponential dependence of the probability on the action. In the case of non dissipative limit, the action is the usual one in mechanics in accordance with the previous result of numerical simulation of random motion. In the case of dissipative motion, the action in the exponent of the exponential probability is just the one proposed in a previous work (Q.A. Wang, R. Wang, arXiv:1201.6309), an action defined for the total system including the moving system and its environment receiving the dissipated energy. In both cases, the result implies that the most probable paths are the paths of least action which, in the limit of vanishing randomness, become the regular paths minimizing the action.

Error Exponent for Multiple-Access Channels: Lower Bounds

The problem of bounding the reliability function of a multiple access channel (MAC) is studied. Two new upper bounds on the error exponent of a two-user discrete memoryless (DM)-MAC are derived. The first bound (sphere packing) is an upper bound on the exponent of the average probability of error and is the first bound of this type that is zero outside the capacity region and thus results in a tighter sphere-packing exponent when compared with the tightest known exponent derived by Haroutunian. The second bound (minimum distance) is an upper bound on the exponent of the maximal (as opposed to average) probability of error. To obtain this bound, first, an upper bound on the minimum Bhattacharyya distance between codeword pairs is derived. For a certain class of two-user DM-MACs, an upper bound on the exponent of maximal probability of error is derived as a consequence of the upper bound on the minimum Bhattacharyya distance. We analytically evaluate the sphere packing bound for uniform composition codes for an additive and nonsymmetric channel and show that it is tight near the boundary of the capacity region, i.e., equal to the random coding lower bound.

Deception With Side Information in Biometric Authentication Systems

In this paper, we study the probability of successful deception of an uncompressed biometric authentication system with side information at the adversary. It represents the scenario where the adversary may have correlated side information, e.g.,~a partial finger print or a DNA sequence of a relative of the legitimate user. We find the optimal exponent of the deception probability by proving both the achievability and the converse. Our proofs are based on the connection between the problem of deception with side information and the rate distortion problem with side information at both the encoder and decoder.

The Impact of Queue Length Information on Buffer Overflow in Parallel Queues

We consider a system consisting of N parallel queues, served by one server. Time is slotted, and the server serves one of the queues in each time slot, according to some scheduling policy. We first characterize the exponent of the buffer overflow probability and the most likely overflow trajectories under the Longest Queue First (LQF) scheduling policy. Under statistically identical arrivals to each queue, we show that the buffer overflow exponents can be simply expressed in terms of the total system occupancy exponent of m parallel queues, for some m ≤ N. We next turn our attention to the rate of queue length information needed to operate a scheduling policy, and its relationship to the buffer overflow exponents. It is known that queue length blind policies such as processor sharing and random scheduling perform worse than the queue aware LQF policy, when it comes to buffer overflow probability. However, we show that the overflow exponent of the LQF policy can be preserved with arbitrarily infrequent queue length updates.

Learning in Hierarchical Social Networks

We study a social network consisting of agents organized as a hierarchical M-ary rooted tree, common in enterprise and military organizational structures. The goal is to aggregate information to solve a binary hypothesis testing problem. Each agent at a leaf of the tree, and only such an agent, makes a direct measurement of the underlying true hypothesis. The leaf agent then makes a decision and sends it to its supervising agent, at the next level of the tree. Each supervising agent aggregates the decisions from the M members of its group, produces a summary message, and sends it to its supervisor at the next level, and so on. Ultimately, the agent at the root of the tree makes an overall decision. We derive upper and lower bounds for the Type I and II error probabilities associated with this decision with respect to the number of leaf agents, which in turn characterize the converge rates of the Type I, Type II, and total error probabilities. We also provide a message-passing scheme involving non-binary message alphabets and characterize the exponent of the error probability with respect to the message alphabet size.

The Source Identification Game: An Information-Theoretic Perspective

We introduce a theoretical framework in which to cast the source identification problem. Thanks to the adoption of a game-theoretic approach, the proposed framework permits us to derive the ultimate achievable performance of the forensic analysis in the presence of an adversary aiming at deceiving it. The asymptotic Nash equilibrium of the source identification game is derived under an assumption on the resources on which the forensic analyst may rely. The payoff at the equilibrium is analyzed, deriving the conditions under which a successful forensic analysis is possible and the error exponent of the false-negative error probability in such a case. The difficulty of deriving a closed-form solution for general instances of the game is alleviated by the introduction of an efficient numerical procedure for the derivation of the optimum attacking strategy. The numerical analysis is applied to a case study to show the kind of information it can provide.

The Spectral Efficiency of Slotted CSMA Ad-Hoc Networks with Directional Antennas

The performance of wireless ad-hoc networks (WANET) is mainly limited by its self-interference. This interference can be mitigated by applying smart access mechanisms and smart antennas. In this paper we analyze the performance of WANETs applying two-phase slotted carrier sense multiple access (CSMA) mechanism and utilizing directional antennas. We present simple expressions that enable a high accuracy evaluation of the network area spectral efficiency (ASE). The results show that directional antennas only affect the ASE through a scaling factor, which depends on the antennas pattern and the channel path loss. The ASE expression also provides a simple and accurate evaluation of the optimal system parameters, including the optimal network density, interference threshold and users' rates. In particular, the ASE gain for a CSMA WANET over an ALOHA WANET is shown to be approximated very well by the exponent of the back-off probability. The accuracy of the results and the usefulness of the optimization procedure are also illustrated through numerical simulations.

Coding Theorems for a $(2,2)$-Threshold Scheme With Detectability of Impersonation Attacks

Coding theorems on a $(2,2)$ -threshold scheme with an opponent are discussed in an asymptotic setup, where the opponent tries to impersonate one of the two participants. A situation is considered where $n$ secrets $S^{n}$ from a memoryless source is blockwisely encoded to two shares and the two shares are decoded to $S^{n}$ with permitting negligible decoding error. We introduce correlation level of the two shares and characterize the minimum attainable rates of the shares and a uniform random number for realizing a $(2, 2)$ -threshold scheme that is secure against the impersonation attack by the opponent. It is shown that if the correlation level between the two shares equals to $\ell \geq 0$ , the minimum attainable rates coincide with $H(S)+\ell $ , where $H(S)$ denotes the entropy of the source, and the maximum attainable exponent of the success probability of the impersonation attack equals to $\ell $ . It is also shown that a simple scheme using an ordinary $(2,2)$ -threshold scheme attains all the bounds as well.