In Proceedings of the 50th ACM Symposium on Theory of Computing (STOC), pages 493–506, 2018. Extended version in Journal of the ACM.
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 $d ≥1/2$, 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 $d=1/2$).
@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_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}
}