Intrinsic stationarity for vector quantization: Foundation of dual quantization

Gilles Pagès (1) , Benedikt Wilbertz (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Mots clés: Quantization; Stationarity; Voronoi tessellation; Delaunay triangulation; Numerical integration

Domaines: Probabilités [math.PR]

Gilles Pagès, Benedikt Wilbertz. Intrinsic stationarity for vector quantization: Foundation of dual quantization. 2010. ⟨hal-00528485v1⟩