Concentrate, concentrate!

Igal Sason and I have just posted to arXiv our tutorial paper “Concentration of Measure Inequalities in Information Theory, Communications and Coding”, which was submitted to Foundations and Trends in Communications and Information Theory. Here is the abstract:

This tutorial article is focused on some of the key modern mathematical tools that are used for the derivation of concentration inequalities, on their links to information theory, and on their various applications to communications and coding.

The first part of this article introduces some classical concentration inequalities for martingales, and it also derives some recent refinements of these inequalities. The power and versatility of the martingale approach is exemplified in the context of binary hypothesis testing, codes defined on graphs and iterative decoding algorithms, and some other aspects that are related to wireless communications and coding.

The second part of this article introduces the entropy method for deriving concentration inequalities for functions of many independent random variables, and it also exhibits its multiple connections to information theory. The basic ingredients of the entropy method are discussed first in conjunction with the closely related topic of logarithmic Sobolev inequalities. This discussion is complemented by a related viewpoint based on probability in metric spaces. This viewpoint centers around the so-called transportation-cost inequalities, whose roots are in information theory. Some representative results on concentration for dependent random variables are briefly summarized, with emphasis on their connections to the entropy method.

Finally, the tutorial addresses several applications of the entropy method and related information-theoretic tools to problems in communications and coding. These include strong converses for several source and channel coding problems, empirical distributions of good channel codes with non-vanishing error probability, and an information-theoretic converse for concentration of measure.

There are already many excellent sources on concentration of measure; what makes ours different is the emphasis on information-theoretic aspects, both in the general theory and in applications. Comments, suggestions, thoughts are very welcome.

ISIT 2011: plenaries, the Shannon lecture

Better late than never, right? Besides, Anand all but made sure that I would blog about it eventually.

I have attended this year’s Shannon lecture and all the plenaries except for Wojtek Szpankowski‘s talk on the information theory of algorithms and combinatorics (I bravely fought the jetlag and lost), but you can watch it here. Here are, shall we say, some impressionistic sketches based on the notes I was taking.

Blackwell’s proof of Wald’s identity

Every once in a while you come across a mathematical argument of such incredible beauty that you feel compelled to tell the whole world about it. This post is about one such gem: David Blackwell’s 1946 proof of Wald’s identity on the expected value of a randomly stopped random walk. In fact, even forty years after the publication of that paper, in a conversation with Morris DeGroot, Blackwell said: “That’s a paper I’m still very proud of. It just gives me pleasant feelings every time I think about it.”

What have the Romans ever done for us?

In Alfréd Rényi‘s Dialogues on Mathematics, Archimedes says this to King Hieron:

… Mathematics rewards only those who are interested in it not only for its rewards but also for itself. Mathematics is like your daughter, Helena, who suspects every time a suitor appears that he is not really in love with her, but is only interested in her because he wants to be the king’s son-in-law. She wants a husband who loves her for her own beauty, wit and charm, and not for the wealth and power he an get by marrying her. Similarly, mathematics reveals its secrets only to those who approach it with pure love, for its own beauty. Of course, those who do this are also rewarded with results of practical importance. But if somebody asks at each step, “What can I get out of this?” he will not get far. You remember I told you that the Romans would never be really successful in applying mathematics. Well, now you can see why: they are too practical.

I couldn’t help but think of this little gem.