TY - JOUR

T1 - On the Best Lattice Quantizers

AU - Agrell, Erik

AU - Allen, Bruce

PY - 2023/6/30

Y1 - 2023/6/30

N2 - 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.

AB - 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.

KW - Dither autocorrelation

KW - laminated lattice

KW - lattice theory

KW - mean square error

KW - moment of inertia

KW - normalized second moment

KW - product lattice

KW - quantization constant

KW - quantization error

KW - vector quantization

KW - Voronoi region

KW - white noise

UR - http://www.scopus.com/inward/record.url?scp=85163543171&partnerID=8YFLogxK

U2 - 10.1109/TIT.2023.3291313

DO - 10.1109/TIT.2023.3291313

M3 - Article

AN - SCOPUS:85163543171

VL - 69

SP - 7650

EP - 7658

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 12

ER -