In most error correction coding (ECC) frameworks, the typical error metric is the bit error rate (BER) which measures the number of bit errors. For this metric, the positions of the bits are not relevant to the decoding, and in many noise models, not relevant to the BER either. In many applications this is unsatisfactory as typically all bits are not equal and have different significance. We look at ECC from a Bayesian perspective and introduce Bayes estimators with general loss functions to take into account the bit significance. We propose ECC schemes that optimize this error metric. As the problem is highly nonlinear, traditional ECC construction techniques are not applicable. Using exhaustive search is cost prohibitive, and thus we use iterative improvement search techniques to find good codebooks. We optimize both general codebooks and linear codes. We provide numerical experiments to show that they can be superior to classical linear block codes such as Hamming codes and decoding methods such as minimum distance decoding.
While ML approaches have been applied to classical ECC code design (e.g. @cite_4 ), it has not been applied to code design with different bit significance. Also, in contrast to @cite_4 , the current approach can be a data-driven approach and extendable to use information extracted from empirical channel data (see Section ).
@cite_4
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In this paper, we study the problem of applying data mining to facilitate the investigation of money laundering crimes (MLCs). We have identified a new paradigm of problems --- that of automatic community generation based on uni-party data, the data in which there is no direct or explicit link information available. Consequently, we have proposed a new methodology for Link Discovery based on Correlation Analysis (LDCA). We have used MLC group model generation as an exemplary application of this problem paradigm, and have focused on this application to develop a specific method of automatic MLC group model generation based on timeline analysis using the LDCA methodology, called CORAL. A prototype of CORAL method has been implemented, and preliminary testing and evaluations based on a real MLC case data are reported. The contributions of this work are: (1) identification of the uni-party data community generation problem paradigm, (2) proposal of a new methodology LDCA to solve for problems in this paradigm, (3) formulation of the MLC group model generation problem as an example of this paradigm, (4) application of the LDCA methodology in developing a specific solution (CORAL) to the MLC group model generation problem, and (5) development, evaluation, and testing of the CORAL prototype in a real MLC case data. Recently, Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) methods have been proposed for scaling up Monte Carlo computations to large data problems. Whilst these approaches have proven useful in many applications, vanilla SG-MCMC might suffer from poor mixing rates when random variables exhibit strong couplings under the target densities or big scale differences. In this study, we propose a novel SG-MCMC method that takes the local geometry into account by using ideas from Quasi-Newton optimization methods. These second order methods directly approximate the inverse Hessian by using a limited history of samples and their gradients. Our method uses dense approximations of the inverse Hessian while keeping the time and memory complexities linear with the dimension of the problem. We provide a formal theoretical analysis where we show that the proposed method is asymptotically unbiased and consistent with the posterior expectations. We illustrate the effectiveness of the approach on both synthetic and real datasets. Our experiments on two challenging applications show that our method achieves fast convergence rates similar to Riemannian approaches while at the same time having low computational requirements similar to diagonal preconditioning approaches. We introduce a simple technique for analyzing the iterative decoder that is broadly applicable to different classes of codes defined over graphs in certain fading as well as additive white Gaussian noise (AWGN) channels. The technique is based on the observation that the extrinsic information from constituent maximum a posteriori (MAP) decoders is well approximated by Gaussian random variables when the inputs to the decoders are Gaussian. The independent Gaussian model implies the existence of an iterative decoder threshold that statistically characterizes the convergence of the iterative decoder. Specifically, the iterative decoder converges to zero probability of error as the number of iterations increases if and only if the channel E sub b N sub 0 exceeds the threshold. Despite the idealization of the model and the simplicity of the analysis technique, the predicted threshold values are in excellent agreement with the waterfall regions observed experimentally in the literature when the codeword lengths are large. Examples are given for parallel concatenated convolutional codes, serially concatenated convolutional codes, and the generalized low-density parity-check (LDPC) codes of Gallager and Cheng-McEliece (1996). Convergence-based design of asymmetric parallel concatenated convolutional codes (PCCC) is also discussed. We consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter p and (b) the process that adds the errors can be described by a sufficiently “simple” circuit. Codes for such channel models are attractive since, like codes for standard adversarial errors, they can handle channels whose true behavior is unknown or varying over time. For two classes of channels, we provide explicit, efficiently encodable decodable codes of optimal rate where only inefficiently decodable codes were previously known. In each case, we provide one encoder decoder that works for every channel in the class. The encoders are randomized, and probabilities are taken over the (local, unknown to the decoder) coins of the encoder and those of the channel. Unique decoding for additive errors: We give the first construction of a polynomial-time encodable decodable code for additive (a.k.a. oblivious) channels that achieve the Shannon capacity 1 − H(p). These are channels that add an arbitrary error vector e ∈ 0, 1 N of weight at most pN to the transmitted word; the vector e can depend on the code but not on the randomness of the encoder or the particular transmitted word. Such channels capture binary symmetric errors and burst errors as special cases. List decoding for polynomial-time channels: For every constant c > 0, we construct codes with optimal rate (arbitrarily close to 1 − H(p)) that efficiently recover a short list containing the correct message with high probability for channels describable by circuits of size at most Nc. Our construction is not fully explicit but rather Monte Carlo (we give an algorithm that, with high probability, produces an encoder decoder pair that works for all time Nc channels). We are not aware of any channel models considered in the information theory literature other than purely adversarial channels, which require more than linear-size circuits to implement. We justify the relaxation to list decoding with an impossibility result showing that, in a large range of parameters (p > 1 4), codes that are uniquely decodable for a modest class of channels (online, memoryless, nonuniform channels) cannot have positive rate.