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On the quantization goodness of polar lattices

Ling Liu, Shanxiang Lyu, Cong Ling, Baoming Bai

TL;DR

The paper proves that polar lattices tailored for lossy compression achieve quantization-goodness, with the NSM approaching $\frac{1}{2\pi e}$ as lattice dimension grows. By combining Construction D lattices with nested polar codes and lattice Gaussian shaping, the authors bound distortion, TV distance, and volume-to-noise ratio, establishing that the NSM converges to the Shannon limit and quantization noise matches the target $\tilde{\sigma}_\Delta^2$ asymptotically. The convergence rate is linked to the scaling properties of polar codes, yielding a finite-length bound of the form $O(\log N / N^{1/\mu})$, where $\mu$ is the polar-code scaling exponent, and the analysis shows that fixing the lattice offset (no dithering) does not degrade performance. Collectively, these results provide a deterministic, practical pathway to provably optimal high-dimensional quantization for Gaussian sources using polar lattices, with shaping and implementation considerations clarified.

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 $\frac{1}{2πe}$ as the dimension of lattices increases. It has been predicted by Zamir et al. \cite{ZamirQZ96} 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 \cite{LingQZ}, 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.

On the quantization goodness of polar lattices

TL;DR

The paper proves that polar lattices tailored for lossy compression achieve quantization-goodness, with the NSM approaching as lattice dimension grows. By combining Construction D lattices with nested polar codes and lattice Gaussian shaping, the authors bound distortion, TV distance, and volume-to-noise ratio, establishing that the NSM converges to the Shannon limit and quantization noise matches the target asymptotically. The convergence rate is linked to the scaling properties of polar codes, yielding a finite-length bound of the form , where is the polar-code scaling exponent, and the analysis shows that fixing the lattice offset (no dithering) does not degrade performance. Collectively, these results provide a deterministic, practical pathway to provably optimal high-dimensional quantization for Gaussian sources using polar lattices, with shaping and implementation considerations clarified.

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 as the dimension of lattices increases. It has been predicted by Zamir et al. \cite{ZamirQZ96} 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 \cite{LingQZ}, 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.
Paper Structure (22 sections, 12 theorems, 69 equations, 5 figures)

This paper contains 22 sections, 12 theorems, 69 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries of Polar Lattices
  3. Polar Codes
  4. Lattice Codes and Polar Lattices
  5. Discrete Gaussian Distribution and Flatness Factor
  6. Polar Lattice Quantizer $\Lambda_Q$
  7. Test Channel with Discrete Gaussian Reconstruction
  8. Performance of The Polar Lattice Quantizer $\Lambda_Q$
  9. The Quantization-goodness of Polar Lattices
  10. Frozen bits do not matter
  11. Conclusion
  12. Appendices
  13. Proof of Proposition \ref{['prop:morelevel']}
  14. Proof of Theorem \ref{['thm:TVlvlr']}
  15. Proof of Lemma \ref{["lem:EPXY'"]}
  16. ...and 7 more sections

Key Result

Lemma 1

Let $\tilde{\sigma}^2_{\Delta}= \frac{\sigma_r^2\Delta}{\sigma_s^2}$. If $\epsilon_{\Lambda}(\tilde{\sigma}_{\Delta}) \leq \frac{1}{2}$, the TV distance between the density $f_{Y'}$ and the Gaussian density $f_Y$ satisfies $\mathbb{V}(f_{Y'},f_{Y}) \leq 2\epsilon_{\Lambda}(\tilde{\sigma}_{\Delta})$.

Figures (5)

  • Figure 1: The genuine test channel (blue) for i.i.d. Gaussian sources and its approximate version (red) with discrete Gaussian reconstruction.
  • Figure 2: The settings of the one-dimensional binary partition chain, where we choose $\frac{1}{\eta^2}= O(N)$, $2^q= \sqrt{N}$, and $r = q -\log(\eta) = O(\log N)$.
  • Figure 3: The lattice quantization processes with (dashed cyan line) and without (solid black line) shaping integrated, which correspond to the quantization rule in our work and that in LingQZ, respectively.
  • Figure 4: The quantization of source $\alpha Y^{[N]}$ with shifted fine lattices. When the shape of $\mathcal{V}(\Lambda_Q)$ is not sharp, as some of its corners being merged in the spherical shell, $\|\alpha y^{[N]}-x^{[N]}\|^2 \approx \|\alpha y^{[N]}-x'^{[N]}\|^2$.
  • Figure 5: A demonstration of $\mathcal{V}(\Lambda_Q)$, $\mathcal{P}(\Lambda_Q)$ and the sphere $\mathcal{B}_{\sqrt{N\cdot\mathsf{E}_{Q}[*] + N \cdot \delta}}$ in the two-dimensional case.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Lemma 1: LingBel13
  • Proposition 1: LingQZ
  • Remark 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Remark 2
  • Theorem 2
  • ...and 12 more