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Optimization and Identification of Lattice Quantizers

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Title: Optimization and Identification of Lattice Quantizers
Authors: Agrell, Erik, 1965, Pook-Kolb, Daniel, Allen, Bruce
Source: IEEE Transactions on Information Theory. 71(8):6490-6501
Subject Terms: mean square error, lattice design, numerical optimization, moment of inertia, theta series, Algorithm, stochastic gradient descent, theta image, vector quantization, normalized second moment, laminated lattice, quantization constant, lattice quantization, Voronoi region
Description: 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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  Data: Optimization and Identification of Lattice Quantizers
– Name: Author
  Label: Authors
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  Data: <searchLink fieldCode="AR" term="%22Agrell%2C+Erik%22">Agrell, Erik</searchLink>, 1965<br /><searchLink fieldCode="AR" term="%22Pook-Kolb%2C+Daniel%22">Pook-Kolb, Daniel</searchLink><br /><searchLink fieldCode="AR" term="%22Allen%2C+Bruce%22">Allen, Bruce</searchLink>
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  Data: <i>IEEE Transactions on Information Theory</i>. 71(8):6490-6501
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  Data: <searchLink fieldCode="DE" term="%22mean+square+error%22">mean square error</searchLink><br /><searchLink fieldCode="DE" term="%22lattice+design%22">lattice design</searchLink><br /><searchLink fieldCode="DE" term="%22numerical+optimization%22">numerical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22moment+of+inertia%22">moment of inertia</searchLink><br /><searchLink fieldCode="DE" term="%22theta+series%22">theta series</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithm%22">Algorithm</searchLink><br /><searchLink fieldCode="DE" term="%22stochastic+gradient+descent%22">stochastic gradient descent</searchLink><br /><searchLink fieldCode="DE" term="%22theta+image%22">theta image</searchLink><br /><searchLink fieldCode="DE" term="%22vector+quantization%22">vector quantization</searchLink><br /><searchLink fieldCode="DE" term="%22normalized+second+moment%22">normalized second moment</searchLink><br /><searchLink fieldCode="DE" term="%22laminated+lattice%22">laminated lattice</searchLink><br /><searchLink fieldCode="DE" term="%22quantization+constant%22">quantization constant</searchLink><br /><searchLink fieldCode="DE" term="%22lattice+quantization%22">lattice quantization</searchLink><br /><searchLink fieldCode="DE" term="%22Voronoi+region%22">Voronoi region</searchLink>
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  Label: Description
  Group: Ab
  Data: 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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      – Type: doi
        Value: 10.1109/TIT.2025.3565218
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 12
        StartPage: 6490
    Subjects:
      – SubjectFull: mean square error
        Type: general
      – SubjectFull: lattice design
        Type: general
      – SubjectFull: numerical optimization
        Type: general
      – SubjectFull: moment of inertia
        Type: general
      – SubjectFull: theta series
        Type: general
      – SubjectFull: Algorithm
        Type: general
      – SubjectFull: stochastic gradient descent
        Type: general
      – SubjectFull: theta image
        Type: general
      – SubjectFull: vector quantization
        Type: general
      – SubjectFull: normalized second moment
        Type: general
      – SubjectFull: laminated lattice
        Type: general
      – SubjectFull: quantization constant
        Type: general
      – SubjectFull: lattice quantization
        Type: general
      – SubjectFull: Voronoi region
        Type: general
    Titles:
      – TitleFull: Optimization and Identification of Lattice Quantizers
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            NameFull: Agrell, Erik
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            NameFull: Pook-Kolb, Daniel
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            NameFull: Allen, Bruce
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              M: 01
              Type: published
              Y: 2025
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              Value: 71
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            – TitleFull: IEEE Transactions on Information Theory
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