Sparse graph codes for compression, sensing, and secrecy

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from student PDF version of thesis.Includes bibliographical references (p. 201-212).Sparse graph codes were first introduced by Gallager over 40 years ago. Over the last two decades, such codes have been the subject of intense research, and capacity approaching sparse graph codes with low complexity encoding and decoding algorithms have been designed for many channels. Motivated by the success of sparse graph codes for channel coding, we explore the use of sparse graph codes for four other problems related to compression, sensing, and security. First, we construct locally encodable and decodable source codes for a simple class of sources. Local encodability refers to the property that when the original source data changes slightly, the compression produced by the source code can be updated easily. Local decodability refers to the property that a single source symbol can be recovered without having to decode the entire source block. Second, we analyze a simple message-passing algorithm for compressed sensing recovery, and show that our algorithm provides a nontrivial f1/f1 guarantee. We also show that very sparse matrices and matrices whose entries must be either 0 or 1 have poor performance with respect to the restricted isometry property for the f2 norm. Third, we analyze the performance of a special class of sparse graph codes, LDPC codes, for the problem of quantizing a uniformly random bit string under Hamming distortion. We show that LDPC codes can come arbitrarily close to the rate-distortion bound using an optimal quantizer. This is a special case of a general result showing a duality between lossy source coding and channel coding-if we ignore computational complexity, then good channel codes are automatically good lossy source codes. We also prove a lower bound on the average degree of vertices in an LDPC code as a function of the gap to the rate-distortion bound. Finally, we construct efficient, capacity-achieving codes for the wiretap channel, a model of communication that allows one to provide information-theoretic, rather than computational, security guarantees. Our main results include the introduction of a new security critertion which is an information-theoretic analog of semantic security, the construction of capacity-achieving codes possessing strong security with nearly linear time encoding and decoding algorithms for any degraded wiretap channel, and the construction of capacity-achieving codes possessing semantic security with linear time encoding and decoding algorithms for erasure wiretap channels. Our analysis relies on a relatively small set of tools. One tool is density evolution, a powerful method for analyzing the behavior of message-passing algorithms on long, random sparse graph codes. Another concept we use extensively is the notion of an expander graph. Expander graphs have powerful properties that allow us to prove adversarial, rather than probabilistic, guarantees for message-passing algorithms. Expander graphs are also useful in the context of the wiretap channel because they provide a method for constructing randomness extractors. Finally, we use several well-known isoperimetric inequalities (Harper's inequality, Azuma's inequality, and the Gaussian Isoperimetric inequality) in our analysis of the duality between lossy source coding and channel coding.by Venkat Bala Chandar.Ph.D

Coding mechanisms for communication and compression : analysis of wireless channels and DNA sequencing

textThis thesis comprises of two related but distinct components: Coding arguments for communication channels and information-theoretic analysis for haplotype assembly. The common thread for both problems is utilizing information and coding theoretic principles in understanding their underlying mechanisms. For the first class of problems, I study two practical challenges that prevent optimal discrete codes utilizing in real communication and compression systems, namely, coding over analog alphabet and fading. In particular, I use an expansion coding scheme to convert the original analog channel coding and source coding problems into a set of independent discrete subproblems. By adopting optimal discrete codes over the expanded levels, this low-complexity coding scheme can approach Shannon limit perfectly or in ratio. Meanwhile, I design a polar coding scheme to deal with the unstable state of fading channels. This novel coding mechanism of hierarchically utilizing different types of polar codes has been proved to be ergodic capacity achievable for several fading systems, without channel state information known at the transmitter. For the second class of problems, I build an information-theoretic view for haplotype assembly. More precisely, the recovery of the target pair of haplotype sequences using short reads is rephrased as the joint source-channel coding problem. Two binary messages, representing haplotypes and chromosome memberships of reads, are encoded and transmitted over a channel with erasures and errors, where the channel model reflects salient features of highthroughput sequencing. The focus is on determining the required number of reads for reliable haplotype reconstruction.Electrical and Computer Engineerin