Linear complexity universal decoding with exponential error probability decay

In this manuscript we consider linear complexity binary linear block encoders and decoders that operate universally with exponential error probability decay. Such scenarios may be relevant in wireless scenarios where probability distributions may not be fully characterized due to the dynamic nature of wireless environments. More specifically, we consider the setting of fixed length-to-fixed length near-lossless data compression of a memoryless binary source of unknown probability distribution as well as the dual setting of communicating on a binary symmetric channel (BSC) with unknown crossover probability. We introduce a new 'min-max distance' metric, analogous to minimum distance, that addresses the universal binary setting and has the same properties as that of minimum distance on BSCs with known crossover probability. The code construction and decoding algorithm are universal extensions of the 'expander codes' framework of Barg and Zemor and have identical complexity and exponential error probability performance

Time-sharing vs. source-splitting in the Slepian-Wolf problem: error exponents analysis

We discuss two approaches for decoding at arbitrary rates in the Slepian-Wolf problem - time sharing and source splitting - both of which rely on constituent vertex decoders. We consider the error exponents for both schemes and conclude that source-splitting is more robust at coding at arbitrary rates, as the error exponent for time-sharing degrades significantly at rates near vertices. As a by-product of our analysis, we exhibit an interesting connection between minimum mean-squared error estimation and error exponents

Towards practical minimum-entropy universal decoding

Minimum-entropy decoding is a universal decoding algorithm used in decoding block compression of discrete memoryless sources as well as block transmission of information across discrete memoryless channels. Extensions can also be applied for multiterminal decoding problems, such as the Slepian-Wolf source coding problem. The 'method of types' has been used to show that there exist linear codes for which minimum-entropy decoders achieve the same error exponent as maximum-likelihood decoders. Since minimum-entropy decoding is NP-hard in general, minimum-entropy decoders have existed primarily in the theory literature. We introduce practical approximation algorithms for minimum-entropy decoding. Our approach, which relies on ideas from linear programming, exploits two key observations. First, the 'method of types' shows that that the number of distinct types grows polynomially in n. Second, recent results in the optimization literature have illustrated polytope projection algorithms with complexity that is a function of the number of vertices of the projected polytope. Combining these two ideas, we leverage recent results on linear programming relaxations for error correcting codes to construct polynomial complexity algorithms for this setting. In the binary case, we explicitly demonstrate linear code constructions that admit provably good performance

Rate-splitting for the deterministic broadcast channel

We show that the deterministic broadcast channel, where a single source transmits to M receivers across a deterministic mechanism, may be reduced, via a rate-splitting transformation, to another (2M−1)-receiver deterministic broadcast channel problem where a successive encoding approach suffices. Analogous to rate-splitting for the multiple access channel and source-splitting for the Slepian-Wolf problem, all achievable rates (including non-vertices) apply. This amounts to significant complexity reduction at the encoder

Disease Prevalence, Disease Incidence, and Mortality in the United States and in England

We find disease incidence and prevalence are both higher among Americans in age groups 55-64 and 70-80 indicating that Americans suffer from higher past cumulative disease risk and experience higher immediate risk of new disease onset compared to the English. In contrast, age specific mortality rates are similar in the two countries with an even higher risk among the English after age 65. Our second aim explains large financial gradients in mortality in the two countries. Among 55-64 year olds, we estimate similar health gradients in income and wealth in both countries, but for 70-80 year old, we find no income gradient in UK. Standard behavioral risk factors (work, marriage, obesity, exercise, and smoking) almost fully explain income gradients among 55-64 years old in both countries and a significant part among Americans 70-80 years old. The most likely explanation of no English income gradient relates to their income benefit system. Below the median, retirement benefits are largely flat and independent of past income and hence past health during the working years. Finally, we report evidence using a long panel of American respondents that their subsequent mortality is not related to large changes in wealth experienced during the prior ten year period.health

Attrition and Health in Ageing Studies: Evidence from ELSA and HRS

In this paper we present results of an investigation into observable characteristics associated with attrition in ELSA and the HRS, with a particular focus on whether attrition is systematically related to health outcomes and socioeconomic status (SES). Investigating the links between health and SES is one of the primary goals of the ELSA and HRS, so attrition correlated with these outcomes is a critical concern. We explored some possible reasons for these differences. Survey maturity, mobility, respondent burden, interviewer quality, and differing sampling methods all fail to account for the gap. Differential respondent incentives may play some role, but the impact of respondent incentive is difficult to test. Apparently, cultural differences between the US and Europe population in agreeing to participate and remain in scientific surveys are a more likely explanation.health, attrition

On the geometry of lambda-symmetries, and PDEs reduction

We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form $\mu$, and we speak of $\mu$-prolongations of vector fields and $\mu$-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions

On some new approaches to practical Slepian-Wolf compression inspired by channel coding

This paper considers the problem, first introduced by Ahlswede and Körner in 1975, of lossless source coding with coded side information. Specifically, let X and Y be two random variables such that X is desired losslessly at the decoder while Y serves as side information. The random variables are encoded independently, and both descriptions are used by the decoder to reconstruct X. Ahlswede and Körner describe the achievable rate region in terms of an auxiliary random variable. This paper gives a partial solution for the optimal auxiliary random variable, thereby describing part of the rate region explicitly in terms of the distribution of X and Y