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We develop a new approach to vector quantization, which guarantees anintrinsic stationarity property that also holds, in contrast to regularquantization, for non-optimal quantization grids. This goal is achieved byreplacing the usual nearest neighbor projection operator for Voronoiquantization by a random splitting operator, which maps the random source tothe vertices of a triangle of $d$-simplex. In the quadratic Euclidean case, itis shown that these triangles or $d$-simplices make up a Delaunay triangulationof the underlying grid. Furthermore, we prove the existence of an optimal gridfor this Delaunay -- or dual -- quantization procedure. We also provide astochastic optimization method to compute such optimal grids, here for higherdimensional uniform and normal distributions. A crucial feature of this newapproach is the fact that it automatically leads to a second order quadratureformula for computing expectations, regardless of the optimality of theunderlying grid.

Intrinsic stationarity for vector quantization: Foundation of dual quantization