On the best lattice quantizers

TL;DR

The paper addresses designing the best lattice quantizers by minimizing the normalized second moment (NSM) and proves that the globally optimal lattice has white quantization error, with covariance ${\mathbf{R}}(\hat{Q}_\Lambda)=\frac{E(\hat{Q}_\Lambda)}{n}\mathbf{I}$. It extends this whiteness result to locally optimal lattices and to optimal products, deriving an explicit scale-factor optimization and showing that a product of locally optimal lattices is white when scaled optimally. It provides a constructive upper bound on NSM via product lattices and lamination, and uses these insights to design improved product lattices in dimensions $n=13$–$48$, achieving NSMs below prior upper bounds in several cases (e.g., $K_{12}\otimes\mathbb{Z}$ at $n=13$). The work yields a comprehensive table of best-known lattice quantizers and highlights that product lattices often yield the best-known NSMs, underscoring the need for analytic and algorithmic methods to reach true optimality. These results advance practical lattice quantizers by delivering tighter NSM benchmarks and actionable design strategies for high-dimensional quantization.

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.

Figures (2)

• Figure 1: The Voronoi region $\Omega$ of the product lattice ${\Lambda_\mathrm{p}} = \Lambda_1 \times \Lambda_2$, where $\Lambda_1$ is the two-dimensional hexagonal lattice $A_2$ and $\Lambda_2$ is the one-dimensional integer lattice ${\mathbb{Z}}$. The origin ${\boldsymbol{0}}$ belongs to all three lattices and is the centroid of all three Voronoi regions. The top and bottom facets of $\Omega$ are shifted copies of $\Omega_1$, and the six vertical edges are shifted copies of $\Omega_2$.
• Figure 2: An example of Theorem \ref{['t:upper']} with $n_1=n_2=1$. Each cell is the decision region of the lattice point it contains, and the shaded cells are the fundamental decision regions. Comparing (a) and (b) shows that the NSMs of ${\hat{Q}}_{\Lambda_\mathrm{p}}$ and ${\tilde{Q}}_\Lambda$ are equal, because their fundamental decision regions are identical. Comparing (b) and (c) shows that ${\tilde{Q}}_\Lambda$ cannot have a smaller NSM than ${\hat{Q}}_\Lambda$, because the lattices are identical and ${\hat{Q}}_\Lambda({\boldsymbol{x}})$ minimizes the quantization error for every ${\boldsymbol{x}}$.

Theorems & Definitions (8)

• Theorem 1: Zamir--Feder zamir96, zamir14book
• Theorem 2
• Proposition 3
• Theorem 4
• Corollary 5
• Lemma 6
• Theorem 7
• Corollary 8