Improved Capacity Upper Bounds for the Discrete-Time Poisson Channel

Cheraghchi, Mahdi; Ribeiro, João

doi:10.1109/isit.2018.8437514

Cited by 6 publications

(13 citation statements)

References 20 publications

(54 reference statements)

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“…We obtain sharp estimates on the functions inside the supremums in (5) and (10) in terms of elementary or standard special functions. Can the supremums themselves (i.e., the capacity upper bounds) be upper bounded in terms of such explicit functions?…”

Section: Discussion and Open Problemsmentioning

confidence: 96%

“…The notion and our techniques can be used to model physical channels studied outside the context of deletiontype channels as well, a notable example being the well-known Poisson channel that is of central importance to optical communications systems [41]. Indeed, a subsequent work by the author [10] successfully applies the techniques developed in this work to obtain improved upper bounds on the capacity of the discrete-time Poisson channel. Furthermore, our contributions in probability theory include motivating novel distributions over non-negative integers and a first study of them, which may be of use in other contexts as well.…”

Section: Cheraghchimentioning

confidence: 99%

“…for the deletion channel. As in the Poisson case, it is desirable to obtain sharp upper bound estimates on the term inside the supremum in (10), which turns out to be a concave function of q, in terms of elementary or common special functions. This is a technical task, and in Section 6.1.2 (in particular, Theorem 24), we obtain sandwiching bounds for the parameters of an inverse binomial distribution in terms of the Lerch transcendent (11), a well-studied generalization of the Riemann zeta function given by (cf.…”

Section: Upper Bounding the Capacity Of General Mean-limited Channelsmentioning

confidence: 99%

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Capacity Upper Bounds for Deletion-type Channels

Cheraghchi

2019

J. ACM

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We develop a systematic approach, based on convex programming and real analysis for obtaining upper bounds on the capacity of the binary deletion channel and, more generally, channels with i.i.d. insertions and deletions. Other than the classical deletion channel, we give special attention to the Poisson-repeat channel introduced by Mitzenmacher and Drinea (IEEE Transactions on Information Theory, 2006). Our framework can be applied to obtain capacity upper bounds for any repetition distribution (the deletion and Poisson-repeat channels corresponding to the special cases of Bernoulli and Poisson distributions). Our techniques essentially reduce the task of proving capacity upper bounds to maximizing a univariate, real-valued, and often concave function over a bounded interval. The corresponding univariate function is carefully designed according to the underlying distribution of repetitions, and the choices vary depending on the desired strength of the upper bounds as well as the desired simplicity of the function (e.g., being only efficiently computable versus having an explicit closed-form expression in terms of elementary, or common special, functions). Among our results, we show the following: (1) The capacity of the binary deletion channel with deletion probability d is at most (1 − d) log φ for d ≥ 1/2 and, assuming that the capacity function is convex, is at most 1 − d log(4/φ) for d < 1/2, where φ = (1 + √ 5)/2 is the golden ratio. This is the first nontrivial capacity upper bound for any value of d outside the limiting case d → 0 that is fully explicit and proved without computer assistance. (2) We derive the first set of capacity upper bounds for the Poisson-repeat channel. Our results uncover further striking connections between this channel and the deletion channel and suggest, somewhat counter-intuitively, that the Poisson-repeat channel is actually analytically simpler than the deletion channel and may be of key importance to a complete understanding of the deletion channel. (3) We derive several novel upper bounds on the capacity of the deletion channel. All upper bounds are maximums of efficiently computable, and concave, univariate real functions over a bounded domain. In turn, we upper bound these functions in terms of explicit elementary and standard special functions, whose maximums can be found even more efficiently (and sometimes analytically, for example, for d = 1/2). CCS Concepts: • Mathematics of computing → Information theory; • Theory of computation → Error-correcting codes;

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Section: Discussion and Open Problemsmentioning

confidence: 96%

Section: Cheraghchimentioning

confidence: 99%

Section: Upper Bounding the Capacity Of General Mean-limited Channelsmentioning

confidence: 99%

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Capacity Upper Bounds for Deletion-type Channels

Cheraghchi

2019

J. ACM

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“…The notion and our techniques can be used to model physical channels studied outside the context of deletion-type channels as well, a notable example being the well-known Poisson channel that is of central importance to optical communications systems [Sha90]. Indeed, a subsequent work by the author [CR18] successfully applies the techniques developed in this work to obtain improved upper bounds on the capacity of the discrete-time Poisson channel. Furthermore, our contributions in probability theory include motivating novel distributions over non-negative integers and a first study of them, which may be of use in other contexts as well.…”

Section: Our Main Contributionsmentioning

confidence: 99%

“…The variation developed in this section can be generally applied to any mean-limited discrete or continuous channel. In a subsequent work by the author[CR18], the technique has been successfully applied to obtain simple and improved upper bounds on the capacity of the discrete-time Poisson channel.…”

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confidence: 99%

Capacity upper bounds for deletion-type channels

Cheraghchi

2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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We develop a systematic approach, based on convex programming and real analysis, for obtaining upper bounds on the capacity of the binary deletion channel and, more generally, channels with i.i.d. insertions and deletions. Other than the classical deletion channel, we give a special attention to the Poisson-repeat channel introduced by Mitzenmacher and Drinea (IEEE Transactions on Information Theory, 2006). Our framework can be applied to obtain capacity upper bounds for any repetition distribution (the deletion and Poisson-repeat channels corresponding to the special cases of Bernoulli and Poisson distributions). Our techniques essentially reduce the task of proving capacity upper bounds to maximizing a univariate, realvalued, and often concave function over a bounded interval. The corresponding univariate function is carefully designed according to the underlying distribution of repetitions and the choices vary depending on the desired strength of the upper bounds as well as the desired simplicity of the function (e.g., being only efficiently computable versus having an explicit closedform expression in terms of elementary, or common special, functions). Among our results, we show the following:1. The capacity of the binary deletion channel with deletion probability d is at most (1 − d) log ϕ for d ≥ 1/2, and, assuming the capacity function is convex, is at most 1−d log(4/ϕ) for d < 1/2, where ϕ = (1 + √ 5)/2 is the golden ratio. This is the first nontrivial capacity upper bound for any value of d outside the limiting case d → 0 that is fully explicit and proved without computer assistance. 2. We derive the first set of capacity upper bounds for the Poisson-repeat channel. Our results uncover further striking connections between this channel and the deletion channel, and suggest, somewhat counter-intuitively, that the Poisson-repeat channel is actually analytically simpler than the deletion channel and may be of key importance to a complete understanding of the deletion channel. 3. We derive several novel upper bounds on the capacity of the deletion channel. All upper bounds are maximums of efficiently computable, and concave, univariate real functions over a bounded domain. In turn, we upper bound these functions in terms of explicit elementary and standard special functions, whose maximums can be found even more efficiently (and sometimes, analytically, for example for d = 1/2).Along the way, we develop several new techniques of potentially independent interest. For example, we develop systematic techniques to study channels with mean constraints over the reals. Furthermore, we motivate the study of novel probability distributions over non-negative integers as well as novel special functions which could be of interest to mathematical analysis. NotationUnless otherwise stated, all logarithms are taken to base e, and the measure of information is converted from nats to bits only for the final numerical estimates. We denote the set of non-negative real numbers by R ≥0 and the set of non-negative integers by N ≥0...

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Amplitude Constrained Poisson Noise Channel: Properties of the Capacity-Achieving Input Distribution

Dytso

Barletta

Shitz

2021

2021 IEEE Information Theory Workshop (ITW)

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This work considers a Poisson noise channel with an amplitude constraint. It is well-known that the capacity-achieving input distribution for this channel is discrete with finitely many points. We sharpen this result by introducing upper and lower bounds on the number of mass points. In particular, the upper bound of order A log 2 (A) and lower bound of order √ A are established where A is the constraint on the input amplitude. In addition, along the way, we show several other properties of the capacity and capacity-achieving distribution. For example, it is shown that the capacity is equal to − log PY (0) where PY is the optimal output distribution. Moreover, an upper bound on the values of the probability masses of the capacity-achieving distribution and a lower bound on the probability of the largest mass point are established.

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Improved Capacity Upper Bounds for the Discrete-Time Poisson Channel

Cited by 6 publications

References 20 publications

Capacity Upper Bounds for Deletion-type Channels

Capacity Upper Bounds for Deletion-type Channels

Capacity upper bounds for deletion-type channels

Amplitude Constrained Poisson Noise Channel: Properties of the Capacity-Achieving Input Distribution

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