English
 
Help Privacy Policy Disclaimer
  Advanced SearchBrowse

Item

ITEM ACTIONSEXPORT
  The optimal lattice quantizer in nine dimensions

Allen, B., & Agrell, E. (2021). The optimal lattice quantizer in nine dimensions. Annalen der Physik, 2021: 2100259. doi:10.1002/andp.202100259.

Item is

Basic

show hide
Genre: Journal Article

Files

show Files
hide Files
:
2104.10107.pdf (Preprint), 434KB
Name:
2104.10107.pdf
Description:
File downloaded from arXiv at 2021-11-16 12:41
Visibility:
Public
MIME-Type / Checksum:
application/pdf / [MD5]
Technical Metadata:
Copyright Date:
-
Copyright Info:
-
:
andp.202100259.pdf (Publisher version), 434KB
Name:
andp.202100259.pdf
Description:
Open Access
Visibility:
Public
MIME-Type / Checksum:
application/pdf / [MD5]
Technical Metadata:
Copyright Date:
-
Copyright Info:
-

Locators

show

Creators

show
hide
 Creators:
Allen, Bruce1, Author              
Agrell , Erik, Author
Affiliations:
1Observational Relativity and Cosmology, AEI-Hannover, MPI for Gravitational Physics, Max Planck Society, ou_24011              

Content

show
hide
Free keywords: Mathematical Physics, math-ph, Astrophysics, Instrumentation and Methods for Astrophysics, astro-ph.IM,General Relativity and Quantum Cosmology, gr-qc,Mathematics, Metric Geometry, math.MG,Mathematics, Mathematical Physics, math.MP
 Abstract: The optimal lattice quantizer is the lattice which minimizes the (dimensionless) second moment $G$. In dimensions $1$ to $8$, it has been proven that the optimal lattice quantizer is one of the classical lattices, or there is good evidence for this. In contrast, more than two decades ago, convincing numerical studies showed that in dimension $9$, a non-classical lattice is optimal. The structure and properties of this lattice depend upon a real parameter $a>0$, whose value was only known approximately. Here, we give a full description of this one-parameter family of lattices and their Voronoi cells, and calculate their (scalar and tensor) second moments analytically as a function of $a$. The value of $a$ which minimizes $G$ is an algebraic number, defined by the root of a $9$th order polynomial, with $a \approx 0.573223794$. 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 $a^2 = 1/2$, $1$ and $2$, where its geometry changes abruptly. At each transition, the analytic formula for the second moment changes in a very simple way. Our methods can be used for arbitrary one-parameter families of layered lattices, and may thus provide a useful tool to identify optimal quantizers in other dimensions as well.

Details

show
hide
Language(s):
 Dates: 2021-04-202021-10-262021
 Publication Status: Published in print
 Pages: Final published version of the paper
 Publishing info: -
 Table of Contents: -
 Rev. Type: -
 Identifiers: arXiv: 2104.10107
DOI: 10.1002/andp.202100259
 Degree: -

Event

show

Legal Case

show

Project information

show

Source 1

show
hide
Title: Annalen der Physik
Source Genre: Journal
 Creator(s):
Affiliations:
Publ. Info: -
Pages: - Volume / Issue: 2021 Sequence Number: 2100259 Start / End Page: - Identifier: -