Optimization and Identification of Lattice Quantizers
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.
Cite
@article{arxiv.2401.01799,
title = {Optimization and Identification of Lattice Quantizers},
author = {Erik Agrell and Daniel Pook-Kolb and Bruce Allen},
journal= {arXiv preprint arXiv:2401.01799},
year = {2025}
}