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On the Best Lattice Quantizers

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Název: On the Best Lattice Quantizers
Autoři: Agrell, Erik, 1965, Allen, Bruce
Zdroj: IEEE Transactions on Information Theory. 69(12):7650-7658
Témata: product lattice, Mean square error methods, quantization constant, Voronoi region, vector quantization, normalized second moment, Symmetric matrices, laminated lattice, quantization error, Dither autocorrelation, Quantization (signal), lattice theory, Lattices, mean square error, Upper bound, moment of inertia, white noise, Generators, Covariance matrices
Popis: A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization error: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any lattice whose mean square error cannot be decreased by a small perturbation of the generator matrix, and (ii) for an optimal product of lattices that are themselves locally optimal in the sense of (i). We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the best known upper bound.
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Přístupová URL adresa: https://research.chalmers.se/publication/536578
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  Data: On the Best Lattice Quantizers
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  Data: <searchLink fieldCode="AR" term="%22Agrell%2C+Erik%22">Agrell, Erik</searchLink>, 1965<br /><searchLink fieldCode="AR" term="%22Allen%2C+Bruce%22">Allen, Bruce</searchLink>
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  Data: <i>IEEE Transactions on Information Theory</i>. 69(12):7650-7658
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  Data: <searchLink fieldCode="DE" term="%22product+lattice%22">product lattice</searchLink><br /><searchLink fieldCode="DE" term="%22Mean+square+error+methods%22">Mean square error methods</searchLink><br /><searchLink fieldCode="DE" term="%22quantization+constant%22">quantization constant</searchLink><br /><searchLink fieldCode="DE" term="%22Voronoi+region%22">Voronoi region</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="%22Symmetric+matrices%22">Symmetric matrices</searchLink><br /><searchLink fieldCode="DE" term="%22laminated+lattice%22">laminated lattice</searchLink><br /><searchLink fieldCode="DE" term="%22quantization+error%22">quantization error</searchLink><br /><searchLink fieldCode="DE" term="%22Dither+autocorrelation%22">Dither autocorrelation</searchLink><br /><searchLink fieldCode="DE" term="%22Quantization+%28signal%29%22">Quantization (signal)</searchLink><br /><searchLink fieldCode="DE" term="%22lattice+theory%22">lattice theory</searchLink><br /><searchLink fieldCode="DE" term="%22Lattices%22">Lattices</searchLink><br /><searchLink fieldCode="DE" term="%22mean+square+error%22">mean square error</searchLink><br /><searchLink fieldCode="DE" term="%22Upper+bound%22">Upper bound</searchLink><br /><searchLink fieldCode="DE" term="%22moment+of+inertia%22">moment of inertia</searchLink><br /><searchLink fieldCode="DE" term="%22white+noise%22">white noise</searchLink><br /><searchLink fieldCode="DE" term="%22Generators%22">Generators</searchLink><br /><searchLink fieldCode="DE" term="%22Covariance+matrices%22">Covariance matrices</searchLink>
– Name: Abstract
  Label: Description
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  Data: A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization error: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any lattice whose mean square error cannot be decreased by a small perturbation of the generator matrix, and (ii) for an optimal product of lattices that are themselves locally optimal in the sense of (i). We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the best known upper bound.
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      – Type: doi
        Value: 10.1109/TIT.2023.3291313
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      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 9
        StartPage: 7650
    Subjects:
      – SubjectFull: product lattice
        Type: general
      – SubjectFull: Mean square error methods
        Type: general
      – SubjectFull: quantization constant
        Type: general
      – SubjectFull: Voronoi region
        Type: general
      – SubjectFull: vector quantization
        Type: general
      – SubjectFull: normalized second moment
        Type: general
      – SubjectFull: Symmetric matrices
        Type: general
      – SubjectFull: laminated lattice
        Type: general
      – SubjectFull: quantization error
        Type: general
      – SubjectFull: Dither autocorrelation
        Type: general
      – SubjectFull: Quantization (signal)
        Type: general
      – SubjectFull: lattice theory
        Type: general
      – SubjectFull: Lattices
        Type: general
      – SubjectFull: mean square error
        Type: general
      – SubjectFull: Upper bound
        Type: general
      – SubjectFull: moment of inertia
        Type: general
      – SubjectFull: white noise
        Type: general
      – SubjectFull: Generators
        Type: general
      – SubjectFull: Covariance matrices
        Type: general
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      – TitleFull: On the Best Lattice Quantizers
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            NameFull: Allen, Bruce
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              Y: 2023
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