Universal polar coding and sparse recovery

TL;DR

This paper develops universal polar coding schemes that adapt to source distributions and applies polarization techniques to sparse signal recovery, achieving near-optimal measurement bounds with low complexity.

Contribution

It introduces a new ordering between distributions for universal polar coding and generalizes decoding algorithms to learn source distributions, enabling universal compression and sparse recovery.

Findings

Universal polar coding operates at the entropy limit with low complexity.

Measurement bounds for sparse recovery match the optimal $O(k \, \log(n/k))$ rate.

Deterministic low complexity measurement matrices are feasible for sparse signals.

Abstract

This paper investigates universal polar coding schemes. In particular, a notion of ordering (called convolutional path) is introduced between probability distributions to determine when a polar compression (or communication) scheme designed for one distribution can also succeed for another one. The original polar decoding algorithm is also generalized to an algorithm allowing to learn information about the source distribution using the idea of checkers. These tools are used to construct a universal compression algorithm for binary sources, operating at the lowest achievable rate (entropy), with low complexity and with guaranteed small error probability. In a second part of the paper, the problem of sketching high dimensional discrete signals which are sparse is approached via the polarization technique. It is shown that the number of measurements required for perfect recovery is competitive with the $O(k \log (n/k))$ bound (with optimal constant for binary signals), meanwhile affording a deterministic low complexity measurement matrix.