General Strong Polarization
Madhu Sudan (Harvard)
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.