Achieving the Fundamental Limit of Lossless Analog Compression via Polarization

TL;DR

This paper introduces a polarization-based framework for lossless analog compression, demonstrating that it achieves the fundamental information-theoretic limit with a deterministic measurement scheme and efficient decoding.

Contribution

It extends polarization phenomena to the analog domain and proposes a practical compression scheme with theoretical optimality and low complexity.

Findings

Polarization of error probability under Hadamard transform for nonsingular sources

Proposed partial Hadamard compression with non-iterative decoding

Achieves the information-theoretic limit for lossless analog compression

Abstract

In this paper, we study the lossless analog compression for i.i.d. nonsingular signals via the polarization-based framework. We prove that for nonsingular source, the error probability of maximum a posteriori (MAP) estimation polarizes under the Hadamard transform, which extends the polarization phenomenon to analog domain. Building on this insight, we propose partial Hadamard compression and develop the corresponding analog successive cancellation (SC) decoder. The proposed scheme consists of deterministic measurement matrices and non-iterative reconstruction algorithm, providing benefits in both space and computational complexity. Using the polarization of error probability, we prove that our approach achieves the information-theoretical limit for lossless analog compression developed by Wu and Verdu.