Intrinsic stationarity for vector quantization: Foundation of dual quantization

TitleIntrinsic stationarity for vector quantization: Foundation of dual quantization
Publication TypeJournal Article
Year of Publication2010
AuthorsGilles Pagès, and Benedikt Wilbertz
KeywordsDelaunay triangulation, numerical integration, quantization, Stationarity, Voronoi tessellation

We develop a new approach to vector quantization, which guarantees an intrinsic stationarity property that also holds, in contrast to regular quantization, for non-optimal 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.

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