EECS500 Spring 2018 Department Colloquium

Mahdi Cherghchi
Capacity Upper Bounds for Deletion-type Channels
Imperial College London
White 411
11:30 AM - 12:00 PM
April 5, 2018
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 $(1-d) \log \varphi$ for $d \geq 1/2$, and, assuming the capacity function is convex, is at most $1-d \log(4/\varphi)$ for $d<1/2$, where $\varphi=(1+\sqrt{5})/2$ 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 $d=1/2$). 
Along the way, we develop several new techniques of potentially independent interest. 
(Based on https://arxiv.org/abs/1711.01630)

Mahdi Cheraghchi is a Lecturer (Assistant Professor) of Computing at Imperial College London and an Adjunct Assistant Professor at the EECS Department at Case Western Reserve University. Previously, he held post-doctoral appointments at UC Berkeley, MIT, CMU, the University of Texas at Austin, as well as a visiting engineer position at Qualcomm. He obtained his PhD in 2010 from EPFL. Mahdi is broadly interested in theoretical computer science, in particular, the role of information and coding theory in cryptography, complexity, algorithms and high-dimensional geometry. He is a senior member of IEEE and ACM.