General Strong Polarization
A martingale is a sequence of random variables that maintain their future expected value conditioned on the past. A $[0,1]$-bounded martingale is said to polarize if it converges in the limit to either $0$ or $1$ with probability $1$. A martingale is said to polarize strongly, if in $t$ steps it is sub-exponentially close to its limit with all but exponentially small probability. In 2008, Arikan built a powerful class of error-correcting codes called Polar codes. The essence of his theory associates a martingale with every invertible square matrix over a field (and a channel) and showed that polarization of the martingale leads to a construction of codes that converge to Shannon capacity. In 2013, Guruswami and Xia, and independently Hassani et al. showed that strong polarization of the Arikan martingale leads to codes that converge to Shannon capacity at finite block lengths, specifically at lengths that are inverse polynomial in the gap to capacity, thereby resolving a major mathematical challenge associated with the attainment of Shannon capacity.
We show that a simple necessary condition for an invertible matrix to polarize over any non-trivial channel is also sufficient for strong polarization over all symmetric channels over all prime fields. Previously the only matrix which was known to polarize strongly was the $2\times 2$ Hadamard matrix. In addition to the generality of our result, it also leads to arguably simpler proofs. The essence of our proof is a “local definition” of polarization which only restricts the evolution of the martingale in a single step, and a general theorem showing the local polarization suffices for strong polarization.
In this talk I will introduce polarization and polar codes and, time permitting, present a full proof of our main theorem. No prior background on polar codes will be assumed. Based on joint work with Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran and Atri Rudra.
Madhu Sudan is a Gordon McKay Professor in the John A. Paulson School of Engineering and Applied Sciences at Harvard University, where he has been since 2015. Madhu Sudan got his Bachelors degree from IIT Delhi in 1987 and his Ph.D. from U.C. Berkeley in 1992. Between 1992 and 2015, Madhu Sudan worked at IBM Research (Research Staff Member 1992-1997), at MIT (Associate Professor 1997-2000, Professor 2000-2011, Fujitsu Chair Professor 2003-2011, CSAIL Associate Director 2007-2009, Adjunct Professor 2011-2015), and at Microsoft Research (Principal Researcher, 2009-2015). Madhu Sudan is a recipient of the Nevanlinna Prize awarded by the International Mathematical Union for outstanding contributions to mathematics of computer and information science, and the Infosys Foundation Prize in Mathematical Sciences. Madhu Sudan is a fellow of the Association for Computing Machinery, the Institute of Electrical and Electronics Engineers and the American Mathematical Society. He is a member of the American Academy of Arts and Sciences and the National Academy of Sciences.
Madhu Sudan’s research interests revolve around mathematical studies of communication and computation. Specifically his research focusses on concepts of reliability and mechanisms that are, or can be, used by computers to interact reliably with each other. His research draws on tools from computational complexity, which studies efficiency of computation, and many areas of mathematics including algebra and probability theory. He is best known for his works on probabilistic checking of proofs, and on the design of list-decoding algorithms for error-correcting codes. His current research interests include property testing which is the study of sublinear time algorithms to estimate properties of massive data, and communication amid uncertainty, a mathematical study of the role of context in communication.