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 nine and ten dimensions with normalized second moments of 0.0716 and 0.0708, respectively, and nonlattice tessellations in seven and nine dimensions with 0.0727 and 0.0711, which improves on previously known values. The new nine- and ten-dimensional lattices suggest that Conway and Sloane's (1993) 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.
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