Details
Original language | English |
---|---|
Article number | 2100259 |
Journal | Annalen der Physik |
Volume | 533 |
Issue number | 12 |
Early online date | 24 Oct 2021 |
Publication status | Published - 14 Dec 2021 |
Abstract
The optimal lattice quantizer is the lattice that minimizes the (dimensionless) second moment G. In dimensions 1 to 3, it has been proven that the optimal lattice quantizer is one of the classical lattices, and there is good numerical evidence for this in dimensions 4 to 8. In contrast, in 9 dimensions, more than two decades ago, the same numerical studies found the smallest known value of G for a non-classical lattice. The structure and properties of this conjectured optimal lattice quantizer depend upon a real parameter (Formula presented.), whose value was only known approximately. Here, a full description of this one-parameter family of lattices and their Voronoi cells is given, and their (scalar and tensor) second moments are calculated analytically as a function of a. The value of a which minimizes G is an algebraic number, defined by the root of a 9th order polynomial, with (Formula presented.). For this value of a, the covariance matrix (second moment tensor) is proportional to the identity, consistent with a theorem of Zamir and Feder for optimal quantizers. The structure of the Voronoi cell depends upon a, and undergoes phase transitions at (Formula presented.), 1, and 2, where its geometry changes abruptly. At each transition, the analytic formula for the second moment changes in a very simple way. The methods can be used for arbitrary one-parameter families of laminated lattices, and may thus provide a useful tool to identify optimal quantizers in other dimensions as well.
Keywords
- lattices, nine dimensions, optimal, quantizers
ASJC Scopus subject areas
- Physics and Astronomy(all)
- General Physics and Astronomy
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In: Annalen der Physik, Vol. 533, No. 12, 2100259, 14.12.2021.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The Optimal Lattice Quantizer in Nine Dimensions
AU - Allen, Bruce
AU - Agrell, Erik
N1 - Funding Information: The authors are grateful for help from the SymPy developers, in particular Oscar Benjamin and Aaron Meurer, in understanding how to automate simplification of expressions involving absolute values, such as (2a2?1)2=|2a2?1|=1?2a2. They also thank Mathieu Dutour Sikiri? for confirming that his face counts for ?9? match those that we found for AE9,[26] and Daniel Pook-Kolb for several helpful comments and?corrections. Open access funding enabled and organized by Projekt DEAL.
PY - 2021/12/14
Y1 - 2021/12/14
N2 - The optimal lattice quantizer is the lattice that minimizes the (dimensionless) second moment G. In dimensions 1 to 3, it has been proven that the optimal lattice quantizer is one of the classical lattices, and there is good numerical evidence for this in dimensions 4 to 8. In contrast, in 9 dimensions, more than two decades ago, the same numerical studies found the smallest known value of G for a non-classical lattice. The structure and properties of this conjectured optimal lattice quantizer depend upon a real parameter (Formula presented.), whose value was only known approximately. Here, a full description of this one-parameter family of lattices and their Voronoi cells is given, and their (scalar and tensor) second moments are calculated analytically as a function of a. The value of a which minimizes G is an algebraic number, defined by the root of a 9th order polynomial, with (Formula presented.). For this value of a, the covariance matrix (second moment tensor) is proportional to the identity, consistent with a theorem of Zamir and Feder for optimal quantizers. The structure of the Voronoi cell depends upon a, and undergoes phase transitions at (Formula presented.), 1, and 2, where its geometry changes abruptly. At each transition, the analytic formula for the second moment changes in a very simple way. The methods can be used for arbitrary one-parameter families of laminated lattices, and may thus provide a useful tool to identify optimal quantizers in other dimensions as well.
AB - The optimal lattice quantizer is the lattice that minimizes the (dimensionless) second moment G. In dimensions 1 to 3, it has been proven that the optimal lattice quantizer is one of the classical lattices, and there is good numerical evidence for this in dimensions 4 to 8. In contrast, in 9 dimensions, more than two decades ago, the same numerical studies found the smallest known value of G for a non-classical lattice. The structure and properties of this conjectured optimal lattice quantizer depend upon a real parameter (Formula presented.), whose value was only known approximately. Here, a full description of this one-parameter family of lattices and their Voronoi cells is given, and their (scalar and tensor) second moments are calculated analytically as a function of a. The value of a which minimizes G is an algebraic number, defined by the root of a 9th order polynomial, with (Formula presented.). For this value of a, the covariance matrix (second moment tensor) is proportional to the identity, consistent with a theorem of Zamir and Feder for optimal quantizers. The structure of the Voronoi cell depends upon a, and undergoes phase transitions at (Formula presented.), 1, and 2, where its geometry changes abruptly. At each transition, the analytic formula for the second moment changes in a very simple way. The methods can be used for arbitrary one-parameter families of laminated lattices, and may thus provide a useful tool to identify optimal quantizers in other dimensions as well.
KW - lattices
KW - nine dimensions
KW - optimal
KW - quantizers
UR - http://www.scopus.com/inward/record.url?scp=85117815517&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2104.10107
DO - 10.48550/arXiv.2104.10107
M3 - Article
AN - SCOPUS:85117815517
VL - 533
JO - Annalen der Physik
JF - Annalen der Physik
SN - 0003-3804
IS - 12
M1 - 2100259
ER -