On the best lattice quantizers

TL;DR

This paper generalizes the properties of optimal lattice quantizers, derives bounds on their normalized second moments, and constructs improved quantizers in multiple dimensions, some of which are the first to beat known bounds.

Contribution

It extends the understanding of optimal lattice quantizers, providing new bounds and methods to construct better quantizers in various dimensions.

Findings

Optimal lattice quantizers have white quantization error.

Derived upper bounds on the normalized second moment of optimal lattices.

Constructed improved lattice quantizers in multiple dimensions, some surpassing previous bounds.

Abstract

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.