Polarization achieved on circular and compact group data sequences
Polarization on Bounded Euclidean Spaces and More General Compact Groups
Information Theory
Summary
The paper studies how a specific technique called polarization works on data sequences that come from circular spaces or more general compact mathematical groups. Polarization means that as data is processed, most parts become nearly deterministic or nearly completely random, which helps in efficient data transmission and compression. The authors show that this effect occurs for memoryless sources on these spaces, but the convergence to polarization is very slow, following a rate related to inverse logarithms. Their findings suggest practical limits on how quickly polarization can be expected to work in such contexts.
What this means in practice
- •For communication system engineers: Design coding schemes for transmitting data modeled on circular or compact group sources using polar codes with known polarization behavior.
- •For signal processing teams: Develop improved algorithms for data compression on data exhibiting circular or group structure informed by slow polarization rates.
A theory result. No direct application yet.
Authors
Sharang M. Sriramu, Lampros Gavalakis, Ioannis Kontoyiannis, Aaron B. Wagner
Abstract
We establish polarization for memoryless sources taking values on the unit-perimeter circle, or on more general compact groups, under Arıkan's polar transform. For sources with finite differential entropy, all but an $O((\log N)^{-1/12})$ fraction of the conditional differential entropies in successive cancellation order lie within any fixed positive distance of zero, where $N$ is the block length. Examples indicate that the true polarization rate is indeed very slow: an inverse-logarithmic rate does not underestimate the polarization speed at moderate block lengths.