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Publication# Adaptive Path Interpolation Method for Sparse Systems: Application to a Censored Block Model

Abstract

Recently, a new adaptive path interpolation method has been developed as a simple and versatile scheme to calculate exactly the asymptotic mutual information of Bayesian inference problems defined on dense factor graphs. These include random linear and generalized estimation, sparse superposition codes, and low-rank matrix / tensor estimation. For all these systems, the adaptive interpolation method directly proves that the replica-symmetric prediction is exact, in a simple and unified manner. When the underlying factor graph of the inference problem is sparse the replica prediction is considerably more complicated, and rigorous results are often lacking or obtained by rather complicated methods. In this work we show how to extend the adaptive path interpolation method to sparse systems. We concentrate on a censored block model, where hidden variables are measured through a binary erasure channel, for which we fully prove the replica prediction.

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Ontological neighbourhood

Multigrid method

In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior. For example, many basic relaxation methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid.

Bayesian inference

Bayesian inference (ˈbeɪziən or ˈbeɪʒən ) is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law.

Statistical inference

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.

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