Correction to 'On the error probability for a class of binary recursive feedback strategies'

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A review on analysis and synthesis of nonlinear stochastic systems with randomly occurring incomplete information

Copyright q 2012 Hongli Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.In the context of systems and control, incomplete information refers to a dynamical system in which knowledge about the system states is limited due to the difficulties in modeling complexity in a quantitative way. The well-known types of incomplete information include parameter uncertainties and norm-bounded nonlinearities. Recently, in response to the development of network technologies, the phenomenon of randomly occurring incomplete information has become more and more prevalent. Such a phenomenon typically appears in a networked environment. Examples include, but are not limited to, randomly occurring uncertainties, randomly occurring nonlinearities, randomly occurring saturation, randomly missing measurements and randomly occurring quantization. Randomly occurring incomplete information, if not properly handled, would seriously deteriorate the performance of a control system. In this paper, we aim to survey some recent advances on the analysis and synthesis problems for nonlinear stochastic systems with randomly occurring incomplete information. The developments of the filtering, control and fault detection problems are systematically reviewed. Latest results on analysis and synthesis of nonlinear stochastic systems are discussed in great detail. In addition, various distributed filtering technologies over sensor networks are highlighted. Finally, some concluding remarks are given and some possible future research directions are pointed out. © 2012 Hongli Dong et al.This work was supported in part by the National Natural Science Foundation of China under Grants 61273156, 61134009, 61273201, 61021002, and 61004067, the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Royal Society of the UK, the National Science Foundation of the USA under Grant No. HRD-1137732, and the Alexander von Humboldt Foundation of German

Optimal Feedback Communication via Posterior Matching

In this paper we introduce a fundamental principle for optimal communication over general memoryless channels in the presence of noiseless feedback, termed posterior matching. Using this principle, we devise a (simple, sequential) generic feedback transmission scheme suitable for a large class of memoryless channels and input distributions, achieving any rate below the corresponding mutual information. This provides a unified framework for optimal feedback communication in which the Horstein scheme (BSC) and the Schalkwijk-Kailath scheme (AWGN channel) are special cases. Thus, as a corollary, we prove that the Horstein scheme indeed attains the BSC capacity, settling a longstanding conjecture. We further provide closed form expressions for the error probability of the scheme over a range of rates, and derive the achievable rates in a mismatch setting where the scheme is designed according to the wrong channel model. Several illustrative examples of the posterior matching scheme for specific channels are given, and the corresponding error probability expressions are evaluated. The proof techniques employed utilize novel relations between information rates and contraction properties of iterated function systems.Comment: IEEE Transactions on Information Theor

Functional Bipartite Ranking: a Wavelet-Based Filtering Approach

It is the main goal of this article to address the bipartite ranking issue from the perspective of functional data analysis (FDA). Given a training set of independent realizations of a (possibly sampled) second-order random function with a (locally) smooth autocorrelation structure and to which a binary label is randomly assigned, the objective is to learn a scoring function s with optimal ROC curve. Based on linear/nonlinear wavelet-based approximations, it is shown how to select compact finite dimensional representations of the input curves adaptively, in order to build accurate ranking rules, using recent advances in the ranking problem for multivariate data with binary feedback. Beyond theoretical considerations, the performance of the learning methods for functional bipartite ranking proposed in this paper are illustrated by numerical experiments

Sequential Necessary and Sufficient Conditions for Capacity Achieving Distributions of Channels with Memory and Feedback

We derive sequential necessary and sufficient conditions for any channel input conditional distribution ${\cal P}_{0,n}\triangleq\{P_{X_t|X^{t-1},Y^{t-1}}:~t=0,\ldots,n\}$ to maximize the finite-time horizon directed information defined by $$C^{FB}_{X^n \rightarrow Y^n} \triangleq \sup_{{\cal P}_{0,n}} I(X^n\rightarrow{Y^n}),~~~ I(X^n \rightarrow Y^n) =\sum_{t=0}^n{I}(X^t;Y_t|Y^{t-1})$$ for channel distributions $\{P_{Y_t|Y^{t-1},X_t}:~t=0,\ldots,n\}$ and $\{P_{Y_t|Y_{t-M}^{t-1},X_t}:~t=0,\ldots,n\}$, where $Y^t\triangleq\{Y_0,\ldots,Y_t\}$ and $X^t\triangleq\{X_0,\ldots,X_t\}$ are the channel input and output random processes, and $M$ is a finite nonnegative integer. \noi We apply the necessary and sufficient conditions to application examples of time-varying channels with memory and we derive recursive closed form expressions of the optimal distributions, which maximize the finite-time horizon directed information. Further, we derive the feedback capacity from the asymptotic properties of the optimal distributions by investigating the limit $$C_{X^\infty \rightarrow Y^\infty}^{FB} \triangleq \lim_{n \longrightarrow \infty} \frac{1}{n+1} C_{X^n \rightarrow Y^n}^{FB}$$ without any \'a priori assumptions, such as, stationarity, ergodicity or irreducibility of the channel distribution. The necessary and sufficient conditions can be easily extended to a variety of channels with memory, beyond the ones considered in this paper.Comment: 57 pages, 9 figures, part of the paper was accepted for publication in the proceedings of the IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain 10-15 July, 2016 (Date of submission of the conference paper: 25/1/2016