On the Best Lattice Quantizers
Journal article, 2023
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.
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
Author
Erik Agrell
Chalmers, Electrical Engineering, Communication, Antennas and Optical Networks
Bruce Allen
Max Planck Society
IEEE Transactions on Information Theory
0018-9448 (ISSN) 1557-9654 (eISSN)
Vol. 69 12 7650-7658Areas of Advance
Information and Communication Technology
Subject Categories
Electrical Engineering, Electronic Engineering, Information Engineering
DOI
10.1109/TIT.2023.3291313