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Register a new userAn Information-Theoretic View of Stochastic Localization
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Added by
Andrea Montanari
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Given a probability measure $mu$ over ${mathbb R}^n$, it is often useful to approximate it by the convex combination of a small number of probability measures, such that each component is close to a product measure. Recently, Ronen Eldan used a stochastic localization argument to prove a general decomposition result of this type. In Eldans theorem, the `number of components is characterized by the entropy of the mixture, and `closeness to product is characterized by the covariance matrix of each component. We present an elementary proof of Eldans theorem which makes use of an information theory (or estimation theory) interpretation. The proof is analogous to the one of an earlier decomposition result known as the `pinning lemma.
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A key practical constraint on the design of Hybrid automatic repeat request (HARQ) schemes is the size of the on-chip buffer that is available at the receiver to store previously received packets. In fact, in modern wireless standards such as LTE and LTE-A, the HARQ buffer size is one of the main drivers of the modem area and power consumption. This has recently highlighted the importance of HARQ buffer management, that is, of the use of buffer-aware transmission schemes and of advanced compression policies for the storage of received data. This work investigates HARQ buffer management by leveraging information-theoretic achievability arguments based on random coding. Specifically, standard HARQ schemes, namely Type-I, Chase Combining and Incremental Redundancy, are first studied under the assumption of a finite-capacity HARQ buffer by considering both coded modulation, via Gaussian signaling, and Bit Interleaved Coded Modulation (BICM). The analysis sheds light on the impact of different compression strategies, namely the conventional compression log-likelihood ratios and the direct digitization of baseband signals, on the throughput. Then, coding strategies based on layered modulation and optimized coding blocklength are investigated, highlighting the benefits of HARQ buffer-aware transmission schemes. The optimization of baseband compression for multiple-antenna links is also studied, demonstrating the optimality of a transform coding approach.
A finite form of de Finettis representation theorem is established using elementary information-theoretic tools: The distribution of the first $k$ random variables in an exchangeable binary vector of length $ngeq k$ is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided.
We generalize a result by Carlen and Cordero-Erausquin on the equivalence between the Brascamp-Lieb inequality and the subadditivity of relative entropy by allowing for random transformations (a broadcast channel). This leads to a unified perspective on several functional inequalities that have been gaining popularity in the context of proving impossibility results. We demonstrate that the information theoretic dual of the Brascamp-Lieb inequality is a convenient setting for proving properties such as data processing, tensorization, convexity and Gaussian optimality. Consequences of the latter include an extension of the Brascamp-Lieb inequality allowing for Gaussian random transformations, the determination of the multivariate Wyner common information for Gaussian sources, and a multivariate version of Nelsons hypercontractivity theorem. Finally we present an information theoretic characterization of a reverse Brascamp-Lieb inequality involving a random transformation (a multiple access channel).
This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored, and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact formulas for quantities that involve expectations of the logarithm of a positive random variable. Here, in the same spirit, we derive an exact integral representation (in one or two dimensions) of the moment of a nonnegative random variable, or the sum of such independent random variables, where the moment order is a general positive noninteger real (also known as fractional moments). The proposed formula is applied to a variety of examples with an information-theoretic motivation, and it is shown how it facilitates their numerical evaluations. In particular, when applied to the calculation of a moment of the sum of a large number, $n$, of nonnegative random variables, it is clear that integration over one or two dimensions, as suggested by our proposed integral representation, is significantly easier than the alternative of integrating over $n$ dimensions, as needed in the direct calculation of the desired moment.
During the last two decades, concentration of measure has been a subject of various exciting developments in convex geometry, functional analysis, statistical physics, high-dimensional statistics, probability theory, information theory, communications and coding theory, computer science, and learning theory. One common theme which emerges in these fields is probabilistic stability: complicated, nonlinear functions of a large number of independent or weakly dependent random variables often tend to concentrate sharply around their expected values. Information theory plays a key role in the derivation of concentration inequalities. Indeed, both the entropy method and the approach based on transportation-cost inequalities are two major information-theoretic paths toward proving concentration. This brief survey is based on a recent monograph of the authors in the Foundations and Trends in Communications and Information Theory (online available at http://arxiv.org/pdf/1212.4663v8.pdf), and a tutorial given by the authors at ISIT 2015. It introduces information theorists to three main techniques for deriving concentration inequalities: the martingale method, the entropy method, and the transportation-cost inequalities. Some applications in information theory, communications, and coding theory are used to illustrate the main ideas.
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