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arXiv · 2401.01799

Optimization and Identification of Lattice Quantizers

Erik Agrell·Daniel Pook-Kolb·Bruce Allen
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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.

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Erik Agrell, Daniel Pook-Kolb, Bruce Allen. 2024-01-03. Optimization and Identification of Lattice Quantizers. https://doi.org/10.1109/tit.2025.3565218

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