Optimality of the Delaunay triangulation in Rd
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Foundations of Quantization for Probability Distributions
Foundations of Quantization for Probability Distributions
Optimal quantizers for Radon random vectors in a Banach space
Journal of Approximation Theory
Quantization Based Filtering Method Using First Order Approximation
SIAM Journal on Numerical Analysis
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We develop a new approach to vector quantization, which guarantees an intrinsic stationarity property that also holds, in contrast to regular quantization, for nonoptimal quantization grids. This goal is achieved by replacing the usual nearest neighbor projection operator for Voronoi quantization by a random splitting operator, which maps the random source to the vertices of a triangle of $d$-simplex. In the quadratic Euclidean case, it is shown that these triangles or $d$-simplices make up a Delaunay triangulation of the underlying grid. Furthermore, we prove the existence of an optimal grid for this Delaunay, or dual, quantization procedure. We also provide a stochastic optimization method to compute such optimal grids, here for higher dimensional uniform and normal distributions. A crucial feature of this new approach is the fact that it automatically leads to a second order quadrature formula for computing expectations, regardless of the optimality of the underlying grid.