Gradient-Based Optimization of Lattice Quantizers

Author:

Erik Agrell, Daniel Pook-Kolb, Bruce Allen

Keyword:

Computer Science, Information Theory, Information Theory (cs.IT), Mathematical Physics (math-ph), Metric Geometry (math.MG)

journal:

date:

2024-01-03 00:00:00

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.

PDF: Gradient-Based Optimization of Lattice Quantizers.pdf