Capacity Upper Bounds for Deletion-type Channels

Capacity Upper Bounds for Deletion-type Channels. Cheraghchi, M. In Proceedings of the 50th ACM Symposium on Theory of Computing (STOC), pages 493–506, 2018. Extended version in Journal of the 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 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, 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 $(1-d) łog ϕ$ for $d ≥ 1/2$, and, assuming that the capacity function is convex, is at most $1-d łog(4/ϕ)$ for $d<1/2$, where $ϕ=(1+\sqrt{5})/2$ is the golden ratio. This is the first nontrivial capacity upper bound for any value of $d$ outside the limiting case $d \to 0$ 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 $d=1/2$).

@INPROCEEDINGS{ref:conf:Che18:del,
  author =	 {Mahdi Cheraghchi},
  title =	 {Capacity Upper Bounds for Deletion-type Channels},
  year =	 2018,
  booktitle =	 {Proceedings of the 50th {ACM Symposium on Theory of
                  Computing (STOC)}},
  pages =	 {493--506},
  note =	 {Extended version in {Journal of the ACM}.},
  doi =		 {10.1145/3188745.3188768},
  url_Link =	 {https://dl.acm.org/doi/10.1145/3188745.3188768},
  url_Paper =	 {https://arxiv.org/abs/1711.01630},
  abstract =	 {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,
                  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
                  $(1-d) \log \phi$ for $d \geq 1/2$, and, assuming
                  that the capacity function is convex, is at most
                  $1-d \log(4/\phi)$ for $d<1/2$, where
                  $\phi=(1+\sqrt{5})/2$ is the golden ratio. This is
                  the first nontrivial capacity upper bound for any
                  value of $d$ outside the limiting case $d \to 0$
                  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 $d=1/2$).  },
  keywords =	 {Coding theory, Synchronization error-correcting
                  codes, Channel coding}
}

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