A classification of finite rank dimension groups by their representations in ordered real vector spaces

Peer reviewedPostprin

Rates of convergence in adaptive universal vector quantization

We consider the problem of adaptive universal quantization. By adaptive quantization we mean quantization for which the delay associated with encoding the jth sample in a sequence of length n is bounded for all n>j. We demonstrate the existence of an adaptive universal quantization algorithm for which any weighted sum of the rate and the expected mean square error converges almost surely and in expectation as O(√(log log n/log n)) to the corresponding weighted sum of the rate and the distortion-rate function at that rate

Variable dimension weighted universal vector quantization and noiseless coding

A new algorithm for variable dimension weighted universal coding is introduced. Combining the multi-codebook system of weighted universal vector quantization (WUVQ), the partitioning technique of variable dimension vector quantization, and the optimal design strategy common to both, variable dimension WUVQ allows mixture sources to be effectively carved into their component subsources, each of which can then be encoded with the codebook best matched to that source. Application of variable dimension WUVQ to a sequence of medical images provides up to 4.8 dB improvement in signal to quantization noise ratio over WUVQ and up to 11 dB improvement over a standard full-search vector quantizer followed by an entropy code. The optimal partitioning technique can likewise be applied with a collection of noiseless codes, as found in weighted universal noiseless coding (WUNC). The resulting algorithm for variable dimension WUNC is also described

Variable-rate source coding theorems for stationary nonergodic sources

For a stationary ergodic source, the source coding theorem and its converse imply that the optimal performance theoretically achievable by a fixed-rate or variable-rate block quantizer is equal to the distortion-rate function, which is defined as the infimum of an expected distortion subject to a mutual information constraint. For a stationary nonergodic source, however, the. Distortion-rate function cannot in general be achieved arbitrarily closely by a fixed-rate block code. We show, though, that for any stationary nonergodic source with a Polish alphabet, the distortion-rate function can be achieved arbitrarily closely by a variable-rate block code. We also show that the distortion-rate function of a stationary nonergodic source has a decomposition as the average of the distortion-rate functions of the source's stationary ergodic components, where the average is taken over points on the component distortion-rate functions having the same slope. These results extend previously known results for finite alphabets

One-pass adaptive universal vector quantization

The authors introduce a one-pass adaptive universal quantization technique for real, bounded alphabet, stationary sources. The algorithm is set on line without any prior knowledge of the statistics of the sources which it might encounter and asymptotically achieves ideal performance on all sources that it sees. The system consists of an encoder and a decoder. At increasing intervals, the encoder refines its codebook using knowledge about incoming data symbols. This codebook is then described to the decoder in the form of updates on the previous codebook. The accuracy to which the codebook is described increases as the number of symbols seen, and thus the accuracy to which the codebook is known, grows

Universal quantization of parametric sources has redundancy k/2 (log n)/n

Rissanen has shown that there exist universal noiseless codes for {Xi} with per-letter rate redundancy as low as k/2 (log n)/n, where n is the blocklength and k is the number of source parameters. We derive an analogous result for universal quantization: for any given La-grange multiplier λ>0, there exist universal fixed-rate and variable-rate quantizers with per-letter Lagrangian redundancy (i.e., distortion redundancy plus λ times the rate redundancy) as low as λk/2 (log n)/n

Universal lossless source coding with the Burrows Wheeler transform

The Burrows Wheeler transform (1994) is a reversible sequence transformation used in a variety of practical lossless source-coding algorithms. In each, the BWT is followed by a lossless source code that attempts to exploit the natural ordering of the BWT coefficients. BWT-based compression schemes are widely touted as low-complexity algorithms giving lossless coding rates better than those of the Ziv-Lempel codes (commonly known as LZ'77 and LZ'78) and almost as good as those achieved by prediction by partial matching (PPM) algorithms. To date, the coding performance claims have been made primarily on the basis of experimental results. This work gives a theoretical evaluation of BWT-based coding. The main results of this theoretical evaluation include: (1) statistical characterizations of the BWT output on both finite strings and sequences of length n → ∞, (2) a variety of very simple new techniques for BWT-based lossless source coding, and (3) proofs of the universality and bounds on the rates of convergence of both new and existing BWT-based codes for finite-memory and stationary ergodic sources. The end result is a theoretical justification and validation of the experimentally derived conclusions: BWT-based lossless source codes achieve universal lossless coding performance that converges to the optimal coding performance more quickly than the rate of convergence observed in Ziv-Lempel style codes and, for some BWT-based codes, within a constant factor of the optimal rate of convergence for finite-memory source

The benefits of coding over routing in a randomized setting

A novel randomized network coding approach for robust, distributed transmission and compression of information in networks is presented, and its advantages over routing-based approaches is demonstrated

Byzantine modification detection in multicast networks using randomized network coding

Distributed randomized network coding, a robust approach to multicasting in distributed network settings, can be extended to provide Byzantine modification detection without the use of cryptographic functions is presented in this paper

Byzantine Modification Detection in Multicast Networks With Random Network Coding

An information-theoretic approach for detecting Byzantine or adversarial modifications in networks employing random linear network coding is described. Each exogenous source packet is augmented with a flexible number of hash symbols that are obtained as a polynomial function of the data symbols. This approach depends only on the adversary not knowing the random coding coefficients of all other packets received by the sink nodes when designing its adversarial packets. We show how the detection probability varies with the overhead (ratio of hash to data symbols), coding field size, and the amount of information unknown to the adversary about the random code