Upper concave envelopes and auxiliary random variables

Nair, Chandra

doi:10.1007/s12572-013-0081-7

Cited by 31 publications

(20 citation statements)

References 16 publications

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“…The function Υ is sometimes called the lower convex envelope of Υ. The following lemma is based on known connections between lower convex envelopes and auxiliary random variables (see [35] for more applications). Proof.…”

Section: A Geometric Interpretation Of the Hc Ribbonmentioning

confidence: 99%

Monotone Measures for Non-Local Correlations

Beigi

Gohari

2015

IEEE Trans. Inform. Theory

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Non-locality is the phenomenon of observing strong correlations among the outcomes of local measurements of a multipartite physical system. No-signaling boxes are the abstract objects for studying non-locality, and wirings are local operations on the space of no-signaling boxes. This means that, no matter how non-local the nature is, the set of physical non-local correlations must be closed under wirings. Then, one approach to identify the non-locality of nature is to characterize closed sets of non-local correlations. Although non-trivial examples of wirings of no-signaling boxes are known, there is no systematic way to study wirings. In particular, given a set of no-signaling boxes, we do not know a general method to prove that it is closed under wirings. In this paper, we propose the first general method to construct such closed sets of non-local correlations. We show that a well-known measure of correlation, called maximal correlation, when appropriately defined for non-local correlations, is monotonically decreasing under wirings. This establishes a conjecture about the impossibility of simulating isotropic boxes from each other, implying the existence of a continuum of closed sets of non-local boxes under wirings. To prove our main result, we introduce some mathematical tools that may be of independent interest: we define a notion of maximal correlation ribbon as a generalization of maximal correlation, and provide a connection between it and a known object called hypercontractivity ribbon; we show that these two ribbons are monotone under wirings too.

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Section: A Geometric Interpretation Of the Hc Ribbonmentioning

confidence: 99%

Monotone Measures for Non-Local Correlations

Beigi

Gohari

2015

IEEE Trans. Inform. Theory

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“…Combining (26) and (27) completes the proof of Corollary 4. Note that r (d) has the following simple interpretation.…”

Section: A the Erasure Distortion Measurementioning

confidence: 53%

A Lower Bound on the Sum Rate of Multiple Description Coding With Symmetric Distortion Constraints

Song

Shao

2014

IEEE Trans. Inform. Theory

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We derive a single-letter lower bound on the minimum sum rate of multiple description coding with symmetric distortion constraints. For the binary uniform source with the erasure distortion measure or Hamming distortion measure, this lower bound can be evaluated with the aid of certain minimax theorems. A similar minimax theorem is established in the quadratic Gaussian setting, which is further leveraged to analyze the special case where the minimum sum rate subject to two levels of distortion constraints (with the second level imposed on the complete set of descriptions) is attained; in particular, we determine the minimum achievable distortions at the intermediate levels.Index Terms-Erasure distortion, Hamming distortion, mean squared error, minimax theorem, multiple description coding, saddle point.

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“…Explicit computation of the tensorizing regions defined so far for a given joint distribution can be computationally cumbersome, specially for distributions defined on large alphabet sets. This computation can be relatively simplified if one observes that expressions with auxiliary random variables generally have alternative representations in terms of lower convex envelopes 3 (see e.g., [16]). Consider for instance…”

Section: Computation Of the Regions And Their Local Perturbationmentioning

confidence: 99%

On the duality of additivity and tensorization

Beigi

Gohari

2015

2015 IEEE International Symposium on Information Theory (ISIT)

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A function is said to be additive if, similar to mutual information, expands by a factor of n, when evaluated on n i.i.d. repetitions of a source or channel. On the other hand, a function is said to satisfy the tensorization property if it remains unchanged when evaluated on i.i.d. repetitions. Additive rate regions are of fundamental importance in network information theory, serving as capacity regions or upper bounds thereof. Tensorizing measures of correlation have also found applications in distributed source and channel coding problems as well as the distribution simulation problem. Prior to our work only two measures of correlation, namely the hypercontractivity ribbon and maximal correlation (and their derivatives), were known to have the tensorization property. In this paper, we provide a general framework to obtain a region with the tensorization property from any additive rate region. We observe that hypercontractivity ribbon indeed comes from the dual of the rate region of the Gray-Wyner source coding problem, and generalize it to the multipartite case. Then we define other measures of correlation with similar properties from other source coding problems. We also present some applications of our results.

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Upper concave envelopes and auxiliary random variables

Cited by 31 publications

References 16 publications

Monotone Measures for Non-Local Correlations

Monotone Measures for Non-Local Correlations

A Lower Bound on the Sum Rate of Multiple Description Coding With Symmetric Distortion Constraints

On the duality of additivity and tensorization

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