Optimization of lattices for quantization

Abstract

A training algorithm for the design of lattices for vector quantization is presented. The algorithm uses a steepest descent method to adjust a generator matrix, in the search for a lattice whose Voronoi regions have minimal normalized second moment. The numerical elements of the found generator matrices are interpreted and translated into exact values. Experiments show that the algorithm is stable, in the sense that several independent runs reach equivalent lattices. The obtained lattices reach as low second moments as the best previously reported lattices, or even lower. Specifically, we report lattices in 9 and 10 dimensions with normalized second moments of 0.0716 and 0.0708, respectively, and nonlattice tessellations in 7 and 9 dimensions with 0.0727 and 0.0711, which improves on previously known values. The new 9and 10-dimensional lattices suggest that Conway and Sloane's conjecture on the duality between the optimal lattices for packing and quantization might be false. A discussion of the application of lattices in vector quantizer design for various sources, uniform and nonuniform, is included.

Complexity and algorithms for computing Voronoi cells of lattices

“…According to [AE98], the lattice D + 10 is conjectured to be the optimal lattice quantizer. Conway and Sloane approximated G(K 12 ) ([CS99, Table 2.3]) using Monte-Carlo integration; our exact computation fits into their bounds.…”

Vector Quantizing Feature Space with a Regular Lattice

“…A square grid may not be the optimal lattice [1] for our feature space, with SIFT-like descriptors, which are typically compared using Euclidean distance (but see also section 2.5). Nevertheless, we prefer this simple lattice structure as it is more intuitive and allows for easy interpretation and exploration of the feature space (see section 3).…”

Recovery of a lattice generator matrix from its gram matrix for feedback and precoding in MIMO

“…y = (Hxt · (Hx) = xHHHHx = xHGx. (4) Consequently, one can state that G induces a quadratic form and is defmite positive because Il yl l > 0 for any x 7: 0 .…”