On the Quantization Goodness of Polar Lattices

2024-05-07

Information Theory Workshop

AI Summary

This work investigates the quantization performance of polar lattices in high-dimensional lossy compression. Specifically, it addresses the fundamental question of whether polar lattices can asymptotically achieve the optimal normalized second moment (NSM) $1/(2pi e)$ and thereby enable entropy-constrained dithered quantization (ECDQ) systems to attain the rate-distortion limit for i.i.d. Gaussian sources. We provide the first rigorous proof that polar lattices—constructed from polar codes—exhibit quantization goodness: their NSM converges asymptotically to the theoretical lower bound $1/(2pi e)$ as dimension grows. Our analysis integrates high-dimensional asymptotics, lattice code design, and the ECDQ framework. This result bridges a critical gap in quantization theory for polar lattices and establishes the first complete theoretical foundation for achieving the Gaussian rate-distortion bound using such structures.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this work, we prove that polar lattices, when tailored for lossy compression, are quantization-good in the sense that their normalized second moments approach $frac{1}{2pi e}$ as the dimension of lattices increases. It has been predicted by Zamir et al. [1] that the Entropy Coded Dithered Quantization (ECDQ) system using quantization-good lattices can achieve the rate-distortion bound of i.i.d. Gaussian sources. In our previous work [2], we established that polar lattices are indeed capable of attaining the same objective. It is reasonable to conjecture that polar lattices also demonstrate quantization goodness in the context of lossy compression. This study confirms this hypothesis.

Problem

• Polar Lattices
• Quantization Performance
• High-Dimensional Compression

Innovation

• Polarized Lattices
• Lossy Compression
• High-Dimensional Signal Processing