Intrinsic Stationarity for Vector Quantization: Foundation of Dual Quantization

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.