Achieving the Fundamental Limit of Lossless Analog Compression via Polarization

TL;DR

This work extends lossless analog compression to the real domain by showing MAP-error polarization under the Hadamard transform for nonsingular sources, enabling a deterministic partial Hadamard encoder. An analog successive cancellation (SC) decoder reconstructs the signal with $O(N\log N)$ complexity, achieving the Wu-Verdu limit when the rate exceeds the Rényi information dimension $d(X)$. The analysis introduces the weighted discrete entropy and a novel entropy-power-inequality variant to establish the polarization rate and absorption properties, and it connects the scheme to polar codes through structured row selection and sequential decoding. Empirically, the partial Hadamard SC scheme demonstrates favorable performance relative to standard baselines in noiseless compressed sensing, while highlighting areas for robustness and computational refinement in practical settings.

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.

Figures (5)

• Figure 1: Polar transform for $N=8$.
• Figure 2: Plot of $P^{\text{MAP}}_e(Y_k|Y^{k-1})$ versus $k=1,2,\dots,2^9$ with $P_X = 0.5\delta_0+0.5\mathcal{N}(0,1)$.
• Figure 3: The tree-like evolution of conditional distributions for $N = 8$.
• Figure 4: Illustration of the two-stages absorption of weighted discrete entropy. The red nodes and blue nodes at the $m$-th layer stand for the $W_m$ with high RID and low RID, respectively. The black nodes at the $n$-th layer represent the $W_n$ with low weighted discrete entropy.
• Figure 5: The BLER (left) and NMSE (right) under different measurement rate. Signal length $N =512$, source distribution $P_X = 0.8\delta_0 + 0.2\mathcal{N}(0,1)$.

Theorems & Definitions (43)

• Definition 2.1: Mixed Representation
• Definition 2.2: RID Renyi1959
• Proposition 2.1
• Definition 2.3: MAP Estimate and Error Probability
• Definition 2.4: Weighted Discrete Entropy
• Proposition 2.2
• Proposition 2.3
• Theorem 3.1: Polarization of Error Probability
• Definition 3.1
• Definition 3.2: RID Process HA2013