Optimization and Identification of Lattice Quantizers
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908461295468544 |
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| author | Agrell, Erik Pook-Kolb, Daniel Allen, Bruce |
| author_facet | Agrell, Erik Pook-Kolb, Daniel Allen, Bruce |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01799 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimization and Identification of Lattice Quantizers Agrell, Erik Pook-Kolb, Daniel Allen, Bruce Information Theory Mathematical Physics Metric Geometry 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. |
| title | Optimization and Identification of Lattice Quantizers |
| topic | Information Theory Mathematical Physics Metric Geometry |
| url | https://arxiv.org/abs/2401.01799 |