The Zero-Undetected-Error Capacity Approaches the Sperner Capacity

Ahlswede, Cai, and Zhang proved that, in the noise-free limit, the zero-undetected-error capacity is lower bounded by the Sperner capacity of the channel graph, and they conjectured equality. Here we derive an upper bound that proves the conjecture.Comment: 8 Pages; added a section on the definition of Sperner capacity; accepted for publication in the IEEE Transactions on Information Theor

Broadcasting on Random Directed Acyclic Graphs

We study a generalization of the well-known model of broadcasting on trees. Consider a directed acyclic graph (DAG) with a unique source vertex $X$, and suppose all other vertices have indegree $d\geq 2$. Let the vertices at distance $k$ from $X$ be called layer $k$. At layer $0$, $X$ is given a random bit. At layer $k\geq 1$, each vertex receives $d$ bits from its parents in layer $k-1$, which are transmitted along independent binary symmetric channel edges, and combines them using a $d$-ary Boolean processing function. The goal is to reconstruct $X$ with probability of error bounded away from $1/2$ using the values of all vertices at an arbitrarily deep layer. This question is closely related to models of reliable computation and storage, and information flow in biological networks. In this paper, we analyze randomly constructed DAGs, for which we show that broadcasting is only possible if the noise level is below a certain degree and function dependent critical threshold. For $d\geq 3$, and random DAGs with layer sizes $\Omega(\log k)$ and majority processing functions, we identify the critical threshold. For $d=2$, we establish a similar result for NAND processing functions. We also prove a partial converse for odd $d\geq 3$ illustrating that the identified thresholds are impossible to improve by selecting different processing functions if the decoder is restricted to using a single vertex. Finally, for any noise level, we construct explicit DAGs (using expander graphs) with bounded degree and layer sizes $\Theta(\log k)$ admitting reconstruction. In particular, we show that such DAGs can be generated in deterministic quasi-polynomial time or randomized polylogarithmic time in the depth. These results portray a doubly-exponential advantage for storing a bit in DAGs compared to trees, where $d=1$ but layer sizes must grow exponentially with depth in order to enable broadcasting.Comment: 33 pages, double column format. arXiv admin note: text overlap with arXiv:1803.0752

Descodificação através de Machine Learning

In recent years, machine learning has become one of the most rapidly expanding technologies in a variety of technological fields. In general, it allows a computer to learn from data without being expressly designed for a particular purpose. This thesis investigates the application of decoding methods inspired by machine learning to linear block codes, such as Reed-Muller (RM) codes.Recentemente, o Machine Learning tornou-se uma das tecnologias em mais rápida expansão numa variedade de campos tecnológicos. Em geral, permite que um computador aprenda com os dados sem ser expressamente concebido para um fim específico. Esta dissertação investiga a aplicação de métodos de descodificação inspirados no Machine Learning a códigos de blocos lineares, tais como os códigos de Reed-Muller

When all information is not created equal

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 191-196).Following Shannon's landmark paper, the classical theoretical framework for communication is based on a simplifying assumption that all information is equally important, thus aiming to provide a uniform protection to all information. However, this homogeneous view of information is not suitable for a variety of modern-day communication scenarios such as wireless and sensor networks, video transmission, interactive systems, and control applications. For example, an emergency alarm from a sensor network needs more protection than other transmitted information. Similarly, the coarse resolution of an image needs better protection than its finer details. For such heterogeneous information, if providing a uniformly high protection level to all parts of the information is infeasible, it is desirable to provide different protection levels based on the importance of those parts. The main objective of this thesis is to extend classical information theory to address this heterogeneous nature of information. Many theoretical tools needed for this are fundamentally different from the conventional homogeneous setting. One key issue is that bits are no more a sufficient measure of information. We develop a general framework for understanding the fundamental limits of transmitting such information, calculate such fundamental limits, and provide optimal architectures for achieving these limits. Our analysis shows that even without sacrificing the data-rate from channel capacity, some crucial parts of information can be protected with exponential reliability. This research would challenge the notion that a set of homogenous bits should necessarily be viewed as a universal interface to the physical layer; this potentially impacts the design of network architectures. This thesis also develops two novel approaches for simplifying such difficult problems in information theory. Our formulations are based on ideas from graphical models and Euclidean geometry and provide canonical examples for network information theory. They provide fresh insights into previously intractable problems as well as generalize previous related results.by Shashibhushan Prataprao Borade.Ph.D

Lecture Notes on Network Information Theory

These lecture notes have been converted to a book titled Network Information Theory published recently by Cambridge University Press. This book provides a significantly expanded exposition of the material in the lecture notes as well as problems and bibliographic notes at the end of each chapter. The authors are currently preparing a set of slides based on the book that will be posted in the second half of 2012. More information about the book can be found at http://www.cambridge.org/9781107008731/. The previous (and obsolete) version of the lecture notes can be found at http://arxiv.org/abs/1001.3404v4/