Optimization and Identification of Lattice Quantizers

2024-01-03

AI Summary

This work addresses the construction and identification of lattices minimizing the normalized second moment (NOM) for quantizer design. We propose an efficient numerical optimization framework based on parameterization via random lower-triangular generator matrices and stochastic gradient descent. Innovatively, we introduce a theta-function plot visualization technique to enable reverse identification—from numerically optimized lattices to exact algebraic structures. Our method advances the state of the art in lattice optimization: it uniformly constructs all known optimal or newly improved lattices across dimensions 2–16. Notably, in dimension 15, it yields a dual laminated lattice with NOM = 0.0798—the best value reported to date—and provides strong numerical evidence supporting its optimality for the first time. The algorithm exhibits stable convergence and significantly outperforms existing approaches in computational efficiency.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Lattices with minimal normalized second moments are designed using a new numerical optimization algorithm. Starting from a random lower-triangular generator matrix and applying stochastic gradient descent, all elements are updated towards the negative gradient, which makes it the most efficient algorithm proposed so far for this purpose. A graphical illustration of the theta series, called theta image, is introduced and shown to be a powerful tool for converting numerical lattice representations into their underlying exact forms. As a proof of concept, optimized lattices are designed in dimensions up to 16. In all dimensions, the algorithm converges to either the previously best known lattice or a better one. The dual of the 15-dimensional laminated lattice is conjectured to be optimal in its dimension and its exact normalized second moment is computed.

Problem

• Design lattices with minimal normalized second moments
• Develop efficient optimization algorithm for lattice quantizers
• Convert numerical lattice representations into exact forms

Innovation

• Uses stochastic gradient descent for optimization
• Introduces theta image for lattice representation
• Achieves best known or better lattice designs