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Gradient Based Method for the Fusion of Lattice Quantizers

Liyuan Zhang, Hanzhong Cao, Jiaheng Li, Minyang Yu

TL;DR

This work tackles high-dimensional lattice quantizers by learning how to fuse low-dimensional lattices through gradient-based transforms. It introduces two methods, HouseHold Transform (orthogonal via Householder reflections) and Matrix Exponential (learned orthogonal components with potential deviation), to optimize lattice generators with respect to the normalized second moment (NSM). Empirical results across dimensions 13–22 show substantial NSM improvements over orthogonal concatenation, with Matrix Exponential delivering the strongest gains in higher dimensions, approaching theoretical bounds in several cases. The approach offers a parameter-efficient, scalable framework for lattice quantization and is supported by code release for reproducibility and further exploration.

Abstract

In practical applications, lattice quantizers leverage discrete lattice points to approximate arbitrary points in the lattice. An effective lattice quantizer significantly enhances both the accuracy and efficiency of these approximations. In the context of high-dimensional lattice quantization, previous work proposed utilizing low-dimensional optimal lattice quantizers and addressed the challenge of determining the optimal length ratio in orthogonal splicing. Notably, it was demonstrated that fixed length ratios and orthogonality yield suboptimal results when combining low-dimensional lattices. Building on this foundation, another approach employed gradient descent to identify optimal lattices, which inspired us to explore the use of neural networks to discover matrices that outperform those obtained from orthogonal splicing methods. We propose two novel approaches to tackle this problem: the Household Algorithm and the Matrix Exp Algorithm. Our results indicate that both the Household Algorithm and the Matrix Exp Algorithm achieve improvements in lattice quantizers across dimensions 13, 15, 17 to 19, 21, and 22. Moreover, the Matrix Exp Algorithm demonstrates superior efficacy in high-dimensional settings.

Gradient Based Method for the Fusion of Lattice Quantizers

TL;DR

This work tackles high-dimensional lattice quantizers by learning how to fuse low-dimensional lattices through gradient-based transforms. It introduces two methods, HouseHold Transform (orthogonal via Householder reflections) and Matrix Exponential (learned orthogonal components with potential deviation), to optimize lattice generators with respect to the normalized second moment (NSM). Empirical results across dimensions 13–22 show substantial NSM improvements over orthogonal concatenation, with Matrix Exponential delivering the strongest gains in higher dimensions, approaching theoretical bounds in several cases. The approach offers a parameter-efficient, scalable framework for lattice quantization and is supported by code release for reproducibility and further exploration.

Abstract

In practical applications, lattice quantizers leverage discrete lattice points to approximate arbitrary points in the lattice. An effective lattice quantizer significantly enhances both the accuracy and efficiency of these approximations. In the context of high-dimensional lattice quantization, previous work proposed utilizing low-dimensional optimal lattice quantizers and addressed the challenge of determining the optimal length ratio in orthogonal splicing. Notably, it was demonstrated that fixed length ratios and orthogonality yield suboptimal results when combining low-dimensional lattices. Building on this foundation, another approach employed gradient descent to identify optimal lattices, which inspired us to explore the use of neural networks to discover matrices that outperform those obtained from orthogonal splicing methods. We propose two novel approaches to tackle this problem: the Household Algorithm and the Matrix Exp Algorithm. Our results indicate that both the Household Algorithm and the Matrix Exp Algorithm achieve improvements in lattice quantizers across dimensions 13, 15, 17 to 19, 21, and 22. Moreover, the Matrix Exp Algorithm demonstrates superior efficacy in high-dimensional settings.

Paper Structure

This paper contains 16 sections, 3 theorems, 15 equations, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Innovation of our work
  4. Theoretical Preparation
  5. Defination
  6. HouseHold Transform
  7. Matrix Exponential
  8. Method
  9. Learnable Symmetry Matrix
  10. HouseHold Transform
  11. Matrix Exponential
  12. Experiments
  13. Training Experiments
  14. Analysis
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 2.1

Any orthogonal matrix of size n × n can be constructed as a product of at most n such reflections.

Theorems & Definitions (6)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof