Universal polar coding and sparse recovery

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.